LIBRARY 

OF  THE 


UNIVERSITY  OF  CALIFORNIA. 


Class 


THE   FLOW    OF   WATER 

A   NEW   THEOBY 

OF 

THE    MOTION    OF    WATER    UNDER    PRESSURE 

AND    IN   OPEN    CONDUITS    AND   ITS 

PRACTICAL  APPLICATION 


BY 


LOUIS  ^CHMEER 

CIVIL   AND   IRRIGATION   ENGINEER 


NEW   YORK: 

D.  VAN  NOSTRAND  COMPANY 

23  MURRAY  AND  27  WARREN  STS. 
1909 


COPYRIGHT,  1909, 

BY 

D.  VAN  NOSTRAND  COMPANY 
NEW  YORK 


Stanhope  ipreaa 

F.    H.   GILSON     COMPANT 
BOSTON.     U.S.A. 


PREFACE. 

THE  present  work  is  the  outcome  of  a  series  of  investigations 
begun  several  years  ago  with  the  object  of  finding  a  simple 
expression  for  the  phenomenon  of  flow  in  irrigation  channels. 

The  author  hopes  that  his  work  will  prove  of  interest  and 
value  to  the  student  and  useful  to  the  practical  engineer. 

He  also  hopes  that  it  will  stimulate  further  research  and  thus 
tend  to  widen  the  field  of  hydraulic  knowledge. 


LOUIS    SCHMEER. 


Los  GATOS,  CALIFORNIA,  October,  1909 

I 


iff 


194610 


NOTATION. 


8,8       = 


LJ 


Velocity  in  feet  per  second. 

The  mean  hydraulic  radius  of  a  conduit. 

Area  of  cross  section. 
Wet  Perimeter. 

Diameter  of  circular  or  semicircular  conduit. 


f  Diameter  of  a  circular  conduit. 
•<  Depth  of  Semi  square  or  semi  circle. 
[  Depth  of  Water  in  a  Channel. 

Slope  of  Water  Surface. 

Head  in  feet. 


Length  of  conduit  in  feet. 

Fall  of  surface  in  feet. 
Distance  in  feet. 

The  variable  coefficient  in  the  formula  v  =  c  \/r  .  s. 

The  coefficient  of  friction,  loss  of  head  per  unit  area  of  surface  at 
unit  velocity. 

A  coefficient  indicating  the  resistance  of  an  impediment  to  flow. 

Coefficients   indicating  the  degree  of  roughness   of   the  wet  peri- 
meter. 

A  coefficient  indicating  the  variation  of  the  coefficient  c  with  the 
velocity  of  flow. 

A  vertical  distance,  a  head  of  water. 
Length  of  a  conduit,  a  horizontal  distance. 
Width  of  surface  of  water. 


TABLE   OF   CONTENTS. 

PAGE 

INTRODUCTION 1 

PRIMARY  LAWS  OF  PRESSURE  AND  FALL 4 

PRIMARY  LAWS  OF  FLUID  FRICTION 8 

DISTRIBUTION  OF  HEAD 12 

DISTRIBUTION  OF  ENERGY 14 

THE  COEFFICIENT  c  IN  THE  FORMULA  v  =  c  Vrs 15 

PRIMARY  DETERMINATION  OF  THE  COEFFICIENT  c 17 

VARIATION  OF  THE  COEFFICIENT  c 

(a)  with  the  roughness  of  the  wet  perimeter 21 

(6)  with  the  velocity  of  flow 24 

MATHEMATICAL  EXPRESSIONS  FOR  THE  VARIATION  OF  THE  COEFFICIENT  c 
WITH  THE  VELOCITY: 

(a)  for  conduits  under  pressure 32 

(6)  for  open  conduits 43 

(c)  for  channels  in  earth 47 

THE  RESISTANCE  DUE  TO  CURVES 55 

THE  RESISTANCES  DUE  TO  ENTRANCES,  ELBOWS,  ETC 57 

RIVETED  CONDUITS 59 

PRACTICAL  APPLICATION  OF  THE  FORMULA 63 

VALUES  OF  a,  THE  COEFFICIENT  OF  VARIATION  OF  c 70 

VALUES  OF  THE  COEFFICIENTS  c  AND /FOR  CONDUITS  UNDER  PRESSURE  .  .  71 

Loss  OF  HEAD  IN  WELDED  CONDUITS 72 

DIAMETERS,  INTERNAL  AREAS,  RADII  AND  THEIR  ROOTS 73 

ROOTS  OF  MEAN  HYDRAULIC  RADII 74 

VALUES  OF  m  AND  K,  THE  COEFFICIENTS  INDICATING  THE  DEGREE  OF 

ROUGHNESS 76 

ALPHABETICAL  LIST  OF  AUTHORITIES 77 

EXPERIMENTAL  DATA 82 

FORMS  OF  SECTIONS  OF  CONDUITS 113 

SEWERS 118 

EXPONENTIAL  EQUATIONS 121 

(a)  for  conduits  under  pressure 124 

(6)  for  sewers 126 

(c)  for  open  conduits 129 

EXPLANATION  OF  THE  USE  OF  THE  TABLES  OF  VELOCITIES  AND  QUANTITIES  136 

SINES  OF  SLOPES  AND  THEIR  ROOTS 143 

v 


vi  TABLE   OF   CONTENTS 

PAGE 

POWERS  OF  DIAMETERS  OF  CIRCULAR  CONDUITS 145 

POWERS  OF  ME  AN.  HYDRAULIC  RADII  OR  OF  DEPTHS  OF  WATER  IN  THE 

'  FORM  OF  SECTION  MOST  FAVORABLE  TO  FLOW 147,  151 

QUANTITIES  OF  DISCHARGE  IN  CUBIC  FEET  PER  SECOND  OF  A  CONDUIT 

ONE  FOOT  IN  DIAMETER. 155 

VELOCITY  OF  FLOW  IN  A  SEMI  SQUARE  1  FOOT  DEEP 159 

DISCHARGE  OF  A  SEMI  SQUARE  1  FOOT  DEEP 163 

WEIR  DISCHARGES: 

(a)  Francis'  formula 167 

(6)  Bazin's  formula 168 

WEIR  FORMULAE 173 

METHODS  OF  MEASUREMENT: 

(a)  loss  of  head , 183 

(6)  discharges 184 

SURFACE,  MEAN  AND  BOTTOM  VELOCITIES 193 

VARIATION  OF  THE  COEFFICIENT  c  WITH  THE  SLOPE 196 

THE  FORMULA  IN  METRIC  MEASURE 205 

ENGLISH  AND  METRIC  EQUIVALENTS 209 

GREATEST  EFFICIENCY  OF  A  CONDUIT  OF  A  GIVEN  DIAMETER  AS  A  TRANS- 
MITTER OF  ENERGY 211 

MOST  ECONOMICAL  DIAMETER  OF  A  CONDUIT  UNDER  PRESSURE 212 


THE   FLOW  OF  WATER. 


INTRODUCTION. 

THERE  is  no  branch  of  the  science  of  physics  on  which  more 
has  been  written  than  on  hydraulics.  The  master  minds  of  the 
last  four  centuries  have  wrestled  with  the  problem  and  thread 
by  thread  they  have  torn  away  the  veil  of  mystery  that  enveloped 
the  phenomenon  of  flow, 

The  universal  mind  of  Leonardo  da  Vinci  (1452—1519), 
painter,  sculptor,  scientist  and  engineer,  was  the  first  to  pierce 
the  darkness  and  although  he  did  not  give  his  thoughts  on  the 
flow  of  water  mathematical  expression,  we  are  to-day,  with  all 
the  knowledge  and  experience  gained  since  his  time,  astounded 
at  his  clear  and  comprehensive  reasoning. 

The  great  Galileo  (1564-1642)  admitted  that  he  had  less 
trouble  in  finding  the  law  of  motion  of  the  planets  millions  of 
miles  away  than  in  discerning  any  law  in  the  motion  of  water 
in  the  stream  flowing  at  his  feet. 

Torricelli  (1608-1644),  inventor  of  the  barometer,  investi- 
gated the  laws  of  falling  bodies  and  found  that  the  velocities 
of  bodies  falling  free  vary  with  the  square  roots  of  the  heights 
fallen  through,  or  with  VTT. 

Huygens  (1629-1695)  first  found  the  numerical  value  of  g, 
the  acceleration  due  to  gravity;  and  following  him  Bernoulli 
was  (in  1738)  able  to  write  the  fundamental  formula  for  the 
velocities  of  bodies  falling  free, 


On  this  general  theoretical  foundation  our  present  system 
of  hydraulics  has  gradually  been  built.     Brahms  (Dyke  and 


2  THE    FLOW   OF   WATER 

other  Hydraulic  Constructions  1753)  made  the  first  step  towards 
a  practical  application  of  the  then  existing  theories  of  motion 
to  the  motion  of  water  flowing  in  a  channel.  He  found  that 
the  motion  of  water  flowing  in  a  channel  is  not  like  the  motion 
of  water  falling  free,  or  that  of  a  body  rolling  down  an  inclined 
plane  continually  accelerated  in  speed,  but  moves  with  a 
uniform  velocity,  and  that  the  resistance  due  to  the  friction  of  a 
fluid  against  the  walls  of  the  conduit  depends  on  the  relation  of 
the  wet  perimeter  to  the  area  of  the  cross-section  or  on  the  mean 
hydraulic  depth. 

Chezy  (in  1776),  gave  the  ideas  of  Brahms  an  elegant  mathe- 
matical expression  by  writing  for  the  velocity  of  flow 


v  =  c  \/r  .  s 

in  which  c  is  a  coefficient,  which  Chezy  assumed  to  be  constant, 
and  r  the  mean  hydraulic  depth.  This  simple  formula  found 
general  application  in  practice  and  is  still  in  use. 

Subsequent  writers  occupied  themselves  chiefly  with  the 
definition  of  variations  of  the  coefficient  c  in  the  formula  pro- 
posed by  Chezy. 

Owing  to  the  researches  of  Coulomb  (1736-1806)  on  the 
resistance  of  fluids  to  slow  motions,  the  variation  of  the 
coefficient  c  with  the  velocity  of  flow  was  the  first  to  be 
recognized  and  Weisbach  and  others  found  expressions  for  this 
variation. 

If  Darcy  was  not  the  first  to  perceive  the  influence  of  the 
degree  of  roughness  of  the  walls  of  a  conduit  on  the  velocity  of 
flow,  he  at  any  rate  was  the  first  who  thoroughly  investigated 
the  subject.  (Mouvement  de  1'eau  dans  les  tuyaux,  Paris,  1851.) 
Beginning  his  investigations  on  flow  in  conduits  under  pressure 
he  extended  them  to  flow  in  open  conduits  and  under  the 
auspices  of  the  government  of  France  constructed  a  special 
test  channel  596.5  meters  (1956.5  feet)  long  and  2  meters  wide. 
This  channel  was  successively  lined  with  materials  possessing 
characteristic  degrees  of  roughness,  the  cross-section  was  given 
various  forms  and  the  bottom  various  slopes.  To  regulate  the 
discharge  two  reservoirs  were  constructed  at  the  head  of  the 


INTRODUCTION  3 

channel  and  the  water  admitted  through  carefully  tested  sharp- 
edged  orifices  20  centimeters  square.  The  experiments  were 
extended  also  to  flow  in  channels  lined  with  masonry  and  to 
flow  in  channels  in  earth. 

Darcy's  work  was  after  his  death  completed  by  Bazin,  his 
successor  in  the  office  of  Chief  Engineer  of  Bridges  and  Roads 
in  France.  Darcy-Bazin's  experiments  were  made  with  the 
utmost  care  and  precision  and  the  tabulated  data  (Darcy- 
Bazin,  Recherches  Hydrauliques,  Paris,  1856)  bear  the  stamp 
of  scientific  exactness  and  truth;  they  are  mines  of  reliable 
information  on  all  matters  relating  to  flow. 

Darcy's  experiments  on  flow  in  pipes  have  since  his  time  been 
supplemented  by  many  others.     Hamilton  Smith  in  California 
carefully   gauged   the    discharge    of    sheet-iron    riveted    pipes 
under  great  pressures,  and  his  data  rank  in  reliability  with 
those  of   Darcy.     Clemens  Herschel   gauged  the   discharge  of  N 
large    steel-riveted    pipes;    Iben    that    of    pipes    coated    with 
tar;  Adams  and  Noble  the  discharge  of  circular  pipes  of  planed    V 
boards. 

Kutter,  a  Swiss  engineer,  extended  the  researches  of  Darcy- 
Bazin  on  flow  in  open  conduits  to  channels  of  greater  slopes 
and  greater  dimensions  and  published  (in  1869)  the  results  of 
his  investigations  under  the  title  "  Versuch  zur  Aufstellung 
einer  allgemeinen  Formel,"  etc. 

Kutter  and  Ganguillet  elaborated  a  general  formula  intended 
to  define  the  variation  of  the  coefficient  c  in  the  formula  of 
Chezy  with  the  mean  hydraulic  radius,  the  degree  of  roughness 
of  the  walls  of  the  channel  and  also  with  the  slope. 

Despite  its  cumbrousness  this  formula  found  universal  appli- 
cation. It  has,  however,  many  defects  and  is  no  longer  regarded 
as  embodying  any  true  law  of  flow. 

Bazin,  in  his  memoir,  "  Etudes  sur  les  mouvements  des  eaux 
dans  les  canaux  decouverts  "  (Annales  des  Fonts  et  Chaussees, 
Faris,  1898),  reviews  the  accumulated  experimental  data  and 
proposes  a  formula  of  great  simplicity.  It  does  not,  however, 
express  the  variation  of  c  with  the  velocity  or  with  the 
slope. 


4  THE   FLOW   OF   WATER 

PRIMARY   LAWS   OF   PRESSURE   AND   FALL. 

A. 

The  physical  laws  relating  to  fluids  at  rest,  which  are  of  interest 
in  their  relation  to  fluid  motion,  are  briefly  as  follows  : 

1.  The  pressure  of  water  on  a  surface  is  proportional  to  the 
depth  below  the  free  surface. 

Let  H  be  the  vertical  distance  of  a  horizontal  plane  below  the 
free  surface, 

G  the  weight  of  one  cubic  foot  of  water  =  62.37  pounds. 

P  the  pressure  in  pounds  per  square  foot, 

then  P  =  GH  =  62.37  H 

and  the  pressure  per  square  inch 


p  =  H  =  0<433  H 

144 

2.  The  pressure  of  water  is  the  same  at  all  points  in  a  hori- 
zontal plane  irrespective  of  the  horizontal  distance  of  any  point 
in  the  plane  from  the  free  surface.     No  matter  what  the  shape 
of  the  vessel  or  the  length  of  the  conduit  may  be  the  pressure  at 
any  point  is  always  proportional  to  the  vertical  distance  below 
the  free  surface. 

At  the  bottom  of  a  stand  pipe  80  feet  below  the  free  surface 
of  the  water  the  pressure  on  the  area  of  a  circle  4  inches  in 
diameter  will  be 

0.433        80.0        42        0.7854  =  435.2  pounds. 

Let  a  4-inch  pipe  5  miles  long  be  connected  with  the  standpipe 
at  any  point  below  the  free  surface,  and  the  end  of  the  pipe  be 
placed  in  the  same  horizontal  plane  as  the  bottom  of  the  stand- 
pipe,  then,  no  matter  how  many  curves  or  elbows  there  may  be 
in  the  length  of  the  conduit,  the  pressure  will  be  as  before,  equal 
to  435.2  pound. 

3.  If  a  pressure  be  applied  to  the  free  surface  of  the  water, 
this  pressure  is  transmitted  equally  and  undiminished  in  all 
directions,  and  to  any  distance,  horizontal  or  vertical. 

Into  the  upper  end  of  a  pipe  1  foot  in  diameter  and  filled 


PRIMARY   LAWS  OF   PRESSURE   AND   FALL  5 

with  water  let  a  piston  be  inserted  and  a  pressure  of  100  pounds 

100 

applied.    Then  a  pressure  equal  to  p-^rr  =  129.5  pounds  per 

0.7  854 

square  foot  will  be  exerted  on  any  square  foot  of  the  inner 
surface  of  the  pipe,  no  matter  how  great  the  distance.  Let  the 
depth  of  the  water  below  the  surface  be  20  feet.  Then  the 
total  pressure  per  square  foot  will  be 

129.5  +  (20  62.37)  =  1403.9  pounds. 

If  Pj  is  the  external  pressure  in  pounds  per  square  foot,  the 
total  pressure  will  be,  for  any  distance  H, 


The  external  pressure  due  to  the  atmosphere  is  equal  to  14.7 
pounds  per  square  inch.     It  is  consequently  equal  to  that  of  a 

14  7 
column  of  water       ^    =  33.9  feet  in  height. 


B 

Torricelli's  fundamental  theorem  for  the  velocity  of  bodies 
falling  free  is  expressed  by  the  equation : 

1.  v   =  gt 

2.  v2  =  2gh 

3.  h  =  }$2 

Or: 

1.  The  speed  of  fall  is  proportional  to  the  time  of  fall. 

2.  The  square  of  the  speed  is  proportional  to  the  distance 
fallen  through. 

3.  The  distance  fallen  through  is  proportional  to  the  square 
of  the  time  of  fall. 

The  velocity  of  fall  in  feet  per  second  is  consequently: 
At  the  end  of  the  first  second  of  fall  equal  to  g  =  32.2  ft. 
At  the  end  of  the  second  second  of  fall  equal  to  2g  =  64.4  ft. 
At  the  end  of  the  tenth  second  of  fall  equal  to  10  g  =  322.0  ft. 


THE   FLOW  OF   WATER 

The  velocity  of  fall  in  feet  per  second  is  equal. 

At  the  end  of  the  first  foot  of  space  fallen  through  to 

VW  =  8-025. 
At  the  end  of  the  second  foot  of  space  fallen  through  to 

V±g  =  n.34. 

At  the  end  of  the  tenth  foot  of  space  fallen  through  to 

=  25.35. 


The  distance  fallen  through  is  equal: 

At  the  end  of  the  first  second  of  the  time  of  fall  to  J  g  =  16.1  ft. 

At  the  end  of  the  second  second  of  the  time  of  fall  to  J  g  22 
=  64.4  ft. 

At  the  end  of  the  tenth  second  of  the  time  of  fall  to  J  g  102 
=  1610.0  ft. 

C. 

The  laws  of  fall  thus  stated  apply  to  any  body,  solid  or  liquid 
falling  free  in  vacuo. 

For  bodies  falling  in  the  atmosphere,  the  resistance  of  the  air 
has  to  be  considered.  This  resistance  is  proportionally  the 
greater,  the  less  the  density  of  the  body.  Disregarding  the 
resistance  of  the  air,  a  jet  of  water  issuing  from  a  well-formed 
orifice  has  a  velocity  proportional  to  the  square  root  of  the 
height  of  the  column  of  water  above  the  centre  of  gravity  of  the 
orifice. 

Let  h  be  the  head  of  water  above  the  centre  of  gravity  of  the 

orifice. 

b  a  coefficient  of  velocity  differing  with  the  nature  of  the 
orifice,  and  the  velocity  of  the  jet  will  be 


v  =  b  V2 

If  the  discharge  is  into  free  space  the  speed  of  the  motion  will 
continue  to  increase  with  the  distance  fallen  through,  and  if 
hl  be  the  vertical  distance  fallen  through  in  the  atmosphere, 
the  water  will  have,  at  the  end  of  its  journey,  acquired  a  velocity 
equal  to 

v  =  b  \/2g(h~+~h^j  nearly. 


PRIMARY  LAWS   OF   PRESSURE   AND   FALL  7 

*        D. 

The  motion  of  a  rigid  body  descending  in  an  inclined  plane 
infinitely  smooth  is  continually  accelerated;  the  law  of  fall  still 
holds,  only  with  this  difference,  that  in  the  equation 


g  is  replaced  by  g  sin  d,  d  being  the  angle  which  the  inclined 
plane  makes  with  the  horizon.  The  kinetic  energy  or  living 
force  aquired  by  a  body  descending  in  a  plane  infinitely  smooth 
is  equal  to 

Wh  or  \  m  v  2 
W 
in  which  m  =  —  =  the  mass  of  the  body.     The  weight  of  the 

\j 

body  W,  divides  into  two  components;  one,  equal  to  W  sin  d 
acts  parallel  to  the  plane  and  produces  motion  ;  the  other,  equal 
to  W  cos  d,  acts  at  right  angles  to  the  plane. 

When  the  frictional  resistance  between  the  plane  and  the 
descending  body  is  considered,  the  force  that  produces  the 
motion  or  W  sin  d  reduces  to  W  sin  d,  —  zW  cos  d,  z  being  a 
coefficient  of  friction. 

The  acceleration  of  motion  continues  as  long  as  W  sin  d  is 

greater  than  zW  cos  d.      If  they  are  equal,  or  if  -  -,,  or  tangent 

d  is  equal  to  z,  the  coefficient  of  friction,  the  motion  will  cease. 

Following  the  laws  of  motion  of  a  rigid  body,  the  motion  of  a 
perfect  fluid  flowing  down  an  inclined  plane  infinitely  smooth 
would  be  continually  accelerated.  Owing,  however,  to  internal 
friction,  to  its  adhesive  qualities,  and  the  friction  of  the  fluid 
against  the  surface  of  the  channel  in  which  it  flows,  water  soon 
spends  its  accelerating  force  and  the  motion  arrives  at  a  state 
of  steadiness  more  or  less  approaching  uniformity. 

The  motion  of  water  is  said  to  be  steady,  when  at  a  given  point 
of  the  cross-section  the  fluid  arrives  with  the  same  velocity  and 
in  the  same  direction. 

The  motion  is  said  to  be  uniform,  if  in  following  a  given 
course  the  mass  of  water  has  a  constant  velocity. 


8  THE   FLOW   OF  WATER 

The  motion  is  said  to  be  varying,  if  in  following  a  given  course 
the  velocity  varies  from  point  to  point. 

In  our  subsequent  discussions  of  flow  we  always  assume  the 
motion  to  be  uniform,  or  conditions  to  be  such  that  there  is  no 
acceleration  of  velocity  with  increase  of  the  distance  fallen 
through,  that  the  accelerating  forces  are  equalized  by  frictional 
resistances  and  that  the  velocity  of  flow  at  any  point  in  a  given 
course  remains  constant  as  long  as  the  slope  remains  constant. 

PRIMARY  LAWS  OF  FLUID  FRICTION. 

A  plane  surface  moving  in  a  still  body  of  water  is  retarded  in 
its  motion  by  a  resistance  due  to  the  friction  of  the  fluid  against 
the  surface. 

The  subject  of  fluid  friction  was  investigated  by  Coulomb 
by  rotating  disks  of  greater  or  lesser  diameters  and  having 
surfaces  of  a  greater  or  lesser  degree  of  roughness  with  more  or 
less  speed  in  a  still  body  of  water,  at  greater  or  lesser  depths,  and 
ascertaining  the  work  done  under  the  various  conditions. 

The  researches  of  Coulomb  were  extended  by  Froude  in  his 
investigations  on  the  resistance  of  the  surfaces  of  ships  (1870- 
1874).  For  the  rotating  disks  of  Coulomb,  Froude  substituted 
sharp-edged  planks  or  metal  plates  of  greater  or  lesser  length 
and  coated  with  various  substances.  These  he  impelled  to  move 
in  a  still  body  of  water  and  ascertained  the  resistance  by  a  suit- 
able device. 

The  laws  deduced  from  experiments  made  by  these  investi- 
gators may  be  summed  up  as  follows: 

1.  The  pressure  existing  in  any  horizontal  plane  below  the 
free  surface  or  in  any  part  of  a  conduit  under  pressure  has  no 
influence  on  the  friction  of  the  fluid  against  a  solid  surface. 
Though  the  pressure  in  pounds  per  unit  area  may  be  much 
greater  in  one  part  of  a  conduit  than  in  another,  the  frictional 
resistance  of  the  area  is  not  thereby  increased.  This  is  demon- 
strated as  follows: 

A  plank  of  suitable  shape  is  immersed  in  a  still  body  of  water 
just  below  the  surface,  impelled  to  move  at  a  certain  constant 


PRIMARY  LAWS   OF   FLUID    FRICTION  9 

speed,  and  the  resistance  to  motion  ascertained.  If  the  plank 
is  subsequently  placed  at  a  greater  depth  and  impelled  to  move 
at  the  same  constant  speed,  it  is  found  that  the  resistance  to 
motion  has  not  been  increased.  If  a  pipe  of  constant  dimensions 
is  resting  on  an  inclined  plane,  it  can  also  be  shown  that  the 
loss  of  head  due  to  the  frictional  resistance  is  for  equal  lengths 
of  the  conduit  the  same  in  the  lower  part  of  the  conduit 
where  the  pressure  is  greatest,  as  in  the  upper  part,  where  it  is 
least. 

2.  The  resistance  to  motion,  due  to  the  friction  of  a  fluid 
against  a  solid  surface,  is  proportional  to  the  area  of  the  surface. 
This  is  demonstrated  as  follows:  A  plank  of  a  certain  length 
and  width  is  impelled  to  move  at  a  certain  constant  speed  in  a 
still  body  of  water  and  the  work  done  in  foot  pounds  noted. 
If  the  width  of  the  plank  is  subsequently  doubled,  thus  doubling 
the  area  of  its  surface,  and  it  is  impelled  to  move  at  the  same 
constant  speed,  it  is  found  that  the  work  done  in  foot  pounds  is 
also  doubled. 

If  water  flows  in  a  pipe  running  full  it  is  found  that  the  amount 
of  head  consumed  in  overcoming  the  resistance  of  the  walls  is 
proportional  to  the  length  of  the  pipe. 

Let  AQ  be  the  area  of  a  surface  in  square  feet :  W  the  weight 
in  pounds  required  to  move  a  plank  in  a  still  body  of  water 
at  a  velocity  of  one  foot  per  second;  /  the  frictional  resistance 
in  pounds  per  square  foot  of  surface 

then  /  =  y , 

^o 

and  the  total  resistance  to  motion  in  pounds  at  any  velocity 

W  =  fA0v*, 

x  being  the  variable  exponent  of  the  power  of  v,  to  which  the 
resistance  is  proportional. 

As  the  frictional  resistance  in  pounds  per  square  foot  for  a 
velocity  of  one  foot  per  second  corresponds  to  an  equal  pressure 
per  square  foot,  the  head  corresponding  to  the  resistance  is 

equal  to  h  =  -77  • 

(JT 


10  THE   FLOW   OF   WATER 


The  head  equal  to  the  resistance  or  -£-,  multiplied  by  20,  the  accel- 

Cr 

eration  due  to  gravity  or 


G 

is  termed  the  coefficient  of  friction  and  denoted  by  z.    As  / 

z  G 
=  - —  the  total  resistance  of  a  surface  in  pounds  is  equal  to 

t7 

W  =  z  GA   V    • 

The  velocity  of  flow  remaining  after  the  frictional  resistance 
is  equalized  acts  through  a  distance  equal  to  v.  The  total  work 
done  in  foot  pounds  in  overcoming  the  frictional  resistance  of  a 
surface  is  consequently: 


3.  The  resistance  to  motion  due  to  the  friction  of  a  fluid 
against  a  solid  surface  is  for  equal  areas  of  the  surface  greater 
for  a  short  than  for  a  long  surface.     This  is  demonstrated  by 
impelling  two  planks  of  equal  areas  but  different  lengths  to  move 
at  equal  constant  speeds  in  a  still  body  of  water.     It  will  be 
found  that  more  power  is  consumed  in  moving   the     shorter 
plank.     There  is  a  resistance  due  to  the  cutting  edge  of  the 
plank,    this    resistance    is    proportionally    more    apparent   the 
shorter  the  plank,  because  the  total  surface  is  proportionally  less. 

At  the  entrance  of  any  kind  of  a  conduit  head  is  consumed  by 
a  resistance  due  to  shock.  For  short  conduits  this  head  is  an 
appreciable  part  of  the  total  head  consumed.  With  increasing 
length  of  the  conduit  the  head  thus  consumed  becomes  pro- 
portionally less  and  less  in  comparison  with  the  total  loss  of  head 
and  becomes  insignificant  for  very  long  conduits. 

4.  The  resistance  to  motion  due  to  the  friction  of  a  fluid 
against  a  solid  surface  is  increased  by  elbows,  curves,  etc. 

Joessel,  experimenting  on  the  resistance  of  ships,  found  the 
resistance  of  oblique  planes  to  be  equal  to 


=    J  °^'     ** J    A    v 

1  0.39  +  0.61  sin.  a         2g 


PRIMARY   LAWS   OF   FLUID   FRICTION 


11 


in  which  /  is  a  coefficient  indicating  the  degree  of  roughness  of 
the  surface,  varying  between  1.1  and  1.7,  d  the  density  of  the 
fluid,  A  the  area  of  the  surface,  a  the  angle  the  plane  makes  with 
the  line  of  motion. 

The  resistance  to  motion  in  conduits  is  proportional  to  the 
angle  of  deflection,  the  radius  of  a  curve  and  its  length. 

5.  The  resistance  to  motion  due  to  the  friction  of  a  fluid 
against  a  solid  surface  varies  with  the  degree  of  roughness  of 
the  surface.  It  increases  rapidly  as  the  roughness  of  the  surface 
increases.  By  impelling  surfaces  coated  with  different  materials 
to  move  in  a  still  body  of  water  Coulomb  found  the  following 
values  of  2,  the  coefficient  of  friction  and  /,  the  resistance  in 
pounds  per  square  ft. 


Description  of  Surface. 

z 

/ 

For  a  varnished  surface                                

00258 

00250 

For  a  planed  and  painted  plank              .                    ... 

00350 

00339 

For  the  surface  of  iron  ships            

00362 

00351 

For  a  new  painted  iron  plate       .            

00489 

00443 

For  a  surface  coated  with  fine  sand    

00418 

00405 

For  a  surface  coated  with  coarse  sand    

00503 

00488 

6.  The  power  of  the  velocity  to  which  the  frictional  resistance 
is  proportional  is  not  constant.  It  varies  with  the  degree  of 
roughness  of  the  surface;  with  the  length  of  the  surface  in  the 
direction  of  motion:  it  is  also  influenced  by  angles,  curves,  etc., 
in  the  surface. 

By  impelling  surfaces  coated  with  various  materials  and  of 
various  lengths  in  the  direction  of  motion  to  move  in  a  still 
body  of  water  Froude  found  the  following  values  of  #,  the 
exponent  of  the  power  of  v  to  which  the  resistance  is  proportional : 


2 

8 

20 

50 

Varnished  surface                                             .    . 

2  0 

1  85 

1  85 

1  83 

Surface  coated  with  paraffin              

1  94 

1  93 

Surface  coated  with  tinfoil     

2  16 

1  99 

1  90 

1  83 

Surface  coated  with  sand    

2  0 

2  0 

2  0 

2  0 

Length  of  Surface  in  Feet. 


12  THE  FLOW   OF   WATER 

DISTRIBUTION  OF  HEAD. 

Water  issuing  from  a  well-formed  orifice  flows  with  a  velocity 
directly  proportional  to  the  square  root  of  the  vertical  distance 
between  the  centre  of  gravity  of  the  orifice  and  the  free  surface, 
and  the  velocity  will  continue  to  increase  if  the  discharge  is  into 
free  space. 

A  stream  of  water  entering  a  conduit  encounters  various 
frictional  resistance  tending  to  equalize  the  accelerating  forces 
and  uniform  motion  ensues.  The  total  head  consumed  in 
producing  this  uniform  motion  may  be  resolved  into  several 
components  : 

1.  Head  consumed  in  producing  the  velocity.  This  is  always 
equal  to 


and  usually  but  a  small  fraction  of  the  total  head. 

2.  Head  consumed  in  overcoming  the  frictional  resistance 
due  to  the  entrance  of  the  conduit.  Let  20  be  a  coefficient 
indicating  the  resistance  due  to  the  entrance  and  the  head  con- 
sumed will  be 


3.  Head  consumed  in  overcoming  the  frictional  resistance  of 
the  wet  perimeter,  or  of  the  walls  of  the  conduit. 

We  have  previously  seen  that  the  energy  expended  in  over- 
coming the  resistance  of  a  surface  is 

v3 
E  =  z^  GAQ   —  foot  pounds. 

*9 

Replacing  A0,  the  area  of  the  surface  by  its  equivalent  P,  the 
wet  perimeter  multiplied  by  L,  the  length  of  the  conduit,  this  is 


~s 

and  since  Q,  the  discharge,  is  equal  to  AI,  the  area  of  the  cross- 
section  multiplied  by  v,  the  velocity, 


DISTRIBUTION  OF  HEAD  13 

P        L 

and  as  "7~  =  "#" 

Ai      li 

E  L    v2 

we  have  -— ^  =z.  -=  —  • 

A  G      1  R    2g 

As  #,  the  total  force  in  foot  pounds,  is  the  product  of  height  of 
fall,  quantity  and  weight  we  have 

E 
~Q~G~    l 

and  consequently 

h     -  z-  — 

4.  Head  consumed  in  overcoming  the  frictional  resistances 
due  to  curves,  elbows,  changes  of  section,  etc. 

If  zn  is  a  coefficient  indicating  the  resistances  due  to  these 
impediments  to  flow,  the  head  consumed  will  be  equal  to 

v2 

*nW 
Summing  up  all  the  components  we  have 

H  =  h  +  hQ  +  hl  +  hn 

jj       v2          vn          Lv2  vn 


-    +  *; 

From  this  we  have  for  the  velocity 

2gH 


This  is  on  the  assumption  that  the  resistance  of  a  surface  is 
proportional  to  the  square  of  the  speed.  We  have  already 
observed,  however,  that  this  is  not  always  the  case;  it  is  in  fact 
an  exception.  But  we  are  not  yet  in  a  position  to  give  the  true 
indexes  of  the  powers  of  v  to  which  the  resistance  is  pro- 
portional. 


14  THE   FLOW   OF  WATER 

DISTRIBUTION   OF    ENERGY. 

A  quantity  of  water,  GQ,  impounded  at  a  vertical  distance,//", 
above  a  horizontal  plane,  possesses  with  reference  to  that  plane, 
a  stored  up  or  potential  energy  equal  to 

QGH. 

If  by  means  of  a  conduit  of  greater  or  lesser  length  the  water 
is  transported  to  the  horizontal  plane  at  the  vertical  distance 
H,  below  the  free  surface  the  stored-up  energy  is  transformed 
into  work.  The  total  stored-up  energy  resolves  into  several 
components. 

Let  the  difference  of  level  between  the  free  surface  and  the 
horizontal  plane  be  80  feet,  the  length  of  the  asphalt-coated 
cast-iron  conduit  transporting  the  water  10,000  feet,  and  its 
diameter  one  foot. 

Assuming  for  Zi  the  average  value  0.00489  we  have  for  the 
velocity  of  flow  from  the  data  given 

f  64.4  .80  1* 


1  +  0.505  +  0.00489      OK 
*-  O.Zo  -• 

or  v  =  5.11  feet  per  second. 

The  discharge  in  cubic  feet  per  second  will  be 

Q  =  5.11  d*  0.7854  =  4.013  cubic  feet. 

The  total  energy  expended  in  transporting  this  quantity  is 
equal  to 

E  =  4.013  .  62.4  .  80  =  20,033  foot  pounds. 

This  total  energy  of  20,033  foot  pounds  is  consumed  as  follows: 

1.  A  quantity  of  work  is  done  in  producing  the  velocity  of 
flow.     This  is  equal  to 

QG  ^-  =4.013  .  62.4?^  =  101.6  foot  pounds. 
2g  64.4 

2.  Another   quantity  of  work  is  done   in  overcoming  the 
resistance  at  the  entrance.    This  is  equal  to 

QGz0^  =  4.013  .  62.4  .  0.505  ?~^  =  51.3 
This  is  on  the  assumption  that  ZQ  =  0.505. 


DISTRIBUTION   OF   ENERGY  15 

3.  The  principal  part  of  the  work  is  done  in  overcoming  the 
frictional  resistance  of  the  interior  surface  of  the  conduit.  This 
is  equal  to 

QGZ    L*L  =  4.013  .  62.4  .  0.00489  ^2  ^i*  =  19?880 
R  2g  0.25     64.4 

foot  pounds. 

The  sum  of  the  several  quantities  of  work  done  in  trans- 
porting 4.013  cubic  feet  of  water  a  vertical  distance  of  80  and  a 
horizontal  distance  of  10;000  feet  is  equal  to 

101.6  +  51.3  +  19,880  =  20,033  foot  pounds, 
or 


Dividing  both  sides  of  the  equation  by  QG  we  have  as  before 
„       v2          v2          Lv2 


The  C  oefficient  C  in  the  Formula  v  =  C  Vr~s. 

Neglecting  the  loss  of  head  due  to  the  velocity,  the  loss  of 
head  due  to  the  frictional  resistance  of  the  entrance,  and  the  loss 
of  head  due  to  the  resistance  of  other  obstructions  to  flow, 
which  severally  or  combined,  form  but  a  small  part  of  the  total 
head  lost  if  the  conduit  is  of  a  length  of  4,000  times  the  mean 
hydraulic  depth  or  the  velocity  not  great,  we  have 

L  v2 

H  =  z1  —  —  as  the  loss  of  head  due  to  the  frictional  resistance 
R  2g 

of  the  walls  of  the  conduit.     From  this  we  have 
v2       HR 


and 


y  —  is  equal 


The  term      —  is  eual  to  the  coefficient  c  first  introduced  into 


C  = 


16  THE   FLOW   OF   WATER 

hydraulic  calculations  by  Chezy,  a  French  engineer  (in  1776). 
On  account  of  its  brevity,  this  term  is  almost  exclusively  used 
to  indicate  the  frictional  resistance  of  long  conduits  of  all 
descriptions. 

AS  *,  =  ?|    :        . 

in  which  /  =  the  frictional  resistance  in  pounds  per  square  foot  of 
surface, 

G  =  the  weight  of  one  cubic  foot  of  water  =  62.4  we 
may  write 

•2£l! 

2gj 
.G 

and  as  -^  =  head  lost  per  unit  area  of  surface  at  unit  velocity, 
we  have  finally 

c~\f^  i 

*  head  lost  per  unit  area  at  unit  velocity. 

Chezy  and  many  of  his  followers  up  to  the  middle  of  the  last 
century  considered  the  coefficient  c  to  be  a  constant.  The 
researches  of  Coulomb,  the  investigations  of  Prony,  Eytelwein, 
Weisbach  and  others,  however,  revealed  the  fact,  that  it  varies 
with  the  velocity  of  flow.  Later  researches  by  Darcy  and 
Darcy-Bazin  brought  to  light  the  astounding  influence  of  the 
degree  of  roughness  of  the  walls  of  a  channel  and  of  the  value  of 
the  mean  hydraulic  radius  on  the  value  of  c.  The  manifold 
variations  of  c  render  the  problem  of  its  exact  valuation  one  of 
great  difficulty.  A  mathematical  expression  embodying  all 
variations  will  necessarily  be  very  complex;  to  be  of  practical 
value,  however,  it  should  be  as  simple  as  possible.  It  is  some- 
what difficult  to  harmonize  great  exactness  and  great  simplicity 
without  making  sacrifices  at  one  end  or  the  other.  On  this 
account  two  expressions  are  often  found  embodying  the  same 
idea  and  rendering  it  with  great  exactitude  or  great  sim- 
plicity. 


DISTRIBUTION   OF  ENERGY  17 

We  will  now  proceed  to  investigate  the  laws  on  which  the 
variation  of  c  depends  and  to  find  suitable  mathematical  expres- 
sions embodying  these  laws. 

I.  Primary  Determination  of  the  Coefficient  c. 

Going  back  to  first  principles  we  may  ask  the  question :  To 
what  power  of  R,  the  mean  hydraulic  radius,  is  the  velocity  of 
flow  proportional?  Using  the  exponential  equation 


gives      x - ;°g  %  -  ;°g  *• 

log  R,  -  log  RQ 

we  find  that  the  value  of  x  is,  in  the  case  of  channels  in  earth, 
such  as  rivers  and  canals  and  with  R  varying  between  1  and  50 
feet  in  the  majority  of  cases  equal  to 

1  2  or  3 


1.333  2.666  4 

For  this  class  of  conduits  we  may  consequently  write: 


in  which  y  is  variable,  differing  with  the  degree  of  roughness  and 
with  the  slope  of  the  conduit.  As  v  =  c  \/rTsTand  R*  =  t/r  Vr~ 
we  have 

C-vW, 

hence  c  increases  directly  with  Vr. 

c 
Column  5,  Table  I,  gives  values  of  y  =  ir=for  conduits  of 

several  degrees  of  roughness.  It  will  be  observed  that  the 
formula  gives  fairly  constant  values  of  y  only  for  large  conduits, 
such  as  rivers  and  canals, 

For  small  conduits  however  y  increases  with  increase  of  R  if 
the  wet  perimeter  be  smooth,  but  decreases  with  increase  of  R 
if  the  contrary  is  the  case.  Applying  the  exponential  equation 
to  other  classes  of  conduits,  the  following  values  of  x,  the  power 
of  R}  to  which  the  velocity  is  proportional  were  found. 


18  THE  FLOW   OF   WATER 

For  a  semi-circular  channel  of  fine  cement  x  =  0.67 

For  a  semi-circular  channel  of  concrete  x  =  0.68 

For  a  rectangular  channel  of  rough  boards  x  =  0.69 

For  a  rectangular  channel  of  rough  masonry  x  =  0.75 

For  a  channel  carrying  coarse  detritus  x  =  1.00 

The  conclusions  to  be  drawn  from  these  data  may  be  summed 
up  as  follows: 

1.  For  rivers  and  canals  the  power  of  R,  to  which  the  velocity 
is  proportional,  is  approximately  equal  to  |. 

2  For  small  channels  the  power  varies  with  the  degree  of 
roughness  of  the  perimeter  a/id  the  form  of  the  cross-section  of 
the  conduit. 

3.  For  small  channels  the  power  of  R  increases  with  increase 
of  roughness. 

4.  For  the  smoothest  class  of  conduits  the  velocity  is  pro- 
portional to   R°'Q7  for  the  very  roughest  to  R1'0.    Hence  the 
rougher  the  wet  perimeter,  the  more  conditions  are  approached 
resembling  those  pertaining  to  flow  in  permeable  strata,  in  which 
instance  the  velocity  is  proportional  to  the  square  of  the  diameter 
of  the  channel. 

5.  No  formula,  based  on  anyone  sjjigle  power  of  R  can  give 
satisfactory  results  when  applied  to  all  classes  of  conduits. 


UNIVERSITY 

OF 


VARIATION   OF  THE   COEFFICIENT   C 

TABLE  I. 


19 


_    c 

Description  of  Conduit. 

R 

1000  S 

V 

t/r 

Sudbury  Conduit.  Smooth  hard  brick  well 

0.5 

0.189 

1  .134 

138 

pointed. 

0.6 

0.189 

1.371 

135 

0.8 

0.189 

1.515 

131 

1.0 

0.189 

1.754 

127 

1.2 

0.189 

1  .948 

124 

1  .4 

0.189 

2.148 

121 

1.6 

0.189 

2.382 

119 

1.8 

0.189 

2.514 

119 

:. 

2.0 

0.189 

2.683 

116 

2.2 

0.189 

2.843 

114 

2.33 

0.189 

2.929 

113 

Semicircular  channel 

lined  with  pebbles  f 

0.454 

1.5 

2.17 

95.1 

to  £  inch  diameter. 

0.546 

1.5 

2.50 

95.3 

0.619 

1.5 

2.69 

92.5 

0.681 

1.5 

2.93 

92.3 

0.731 

1.5 

3.05 

91.3 

0.784 

1.5 

3.22 

90.4 

0.826 

1  .5 

3.33 

88.4 

0.900 

1  .5 

3.54 

87.6 

0.968 

1.5 

3.73 

85.8 

1.012 

1.5 

3  .95 

87.9 

Solani  Embankment. 

6.32 

0.140 

2.63 

55.9 

Jaoli  Site. 

6.53 

0.144 

2.70 

55.0 

Sides  of  brick  set  in  mud,  bottoms  very 

6.79 

0.145 

2.80 

54.9 

rough. 

7.05 

0.146 

2.81 

53.7 

7.46 

0.160 

2.94 

51  .5 

Linth    Canal,    channel    in    earth,    fairly 

5.14 

0.29 

3.414 

58.6 

regular. 

5.93 

0.30 

3.830 

58.2 

6.48 

0.31 

4.152 

58.2 

7.12 

0.32 

4.418 

56.5 

7.52 

0.33 

4.753 

57.4 

8.09 

0.34 

4.920 

55.8 

8.28 

0.34 

5.058 

56.1 

8.62 

0.35 

5.225 

57.7 

8.87 

0.36 

5.392 

55.5 

9.18 

0.37 

5.530 

54.5 

River  Seine  at  Paris. 

9.48 

0.14 

3.37 

53.1 

10.92 

0.14 

3.74 

52.8 

12.19 

0.14 

3.81 

49.6 

14.50 

0.14 

4.23 

49.4 

15.02 

0.14 

5.11 

49.8 

15.93 

0.14 

4.68 

49.5 

16.85 

0.131 

4.80 

48.6 

18.39 

0.103 

4.69 

51  .8 

Mill  race  at  Pricbam, 

Hungary. 

0.316 

2.2 

0.389 

20.0 

0.336 

2.2 

0.588 

28.4 

Irregular     channel 

lined     with     rubble 

0.442 

2.2 

0.953 

35.7 

masonry. 

0.548 

2.2 

1.135 

37.7 

0.560 

2.2 

1.190 

39.1 

0.566 

2.2 

1.270 

41  .3 

20 


THE   FLOW   OF   WATER 


TABLE  II. 


Description  of  Conduit. 

V 

c 

a 

I 

New      straight      asphalt-coated      wrought-iron 
riveted  pipe  with  screw  joints. 

0.328 
1.171 
3.117 

76.7 
99.9 

108.4 

0.80 
1  .04 
1  .13 

6.148 

117.1 

1  .219 

m  =  0.94 
R  =  0.0677 

10  .535 
12  .786 

124.0 
124.3 

1  .289 
1.291 

II 

2.78 

139.1 

1.122 

Test  pipe  of  clear  cement. 

3.65 
4.20 

139.2 
139.5 

1.139 
1  .140 

m  =  0.95 

4.72 

140.4 

1  .141 

R  =  0.658 

4.79 

141  .2 

.155 

4.92 

141  .4 

.157 

5.81 

141  .4 

.157 

6.58 

142.5 

.166 

III 

1.0 

101.2 

.0 

New  cylinder  joint  asphalt-coated  steel-riveted 
pipe  with  many  curves. 

2.0 
3.0 
3.5 

108.3 
112.8 
113.4 

.09 
.113 
.119 

m  =  0.53 

4.0 

113.2 

.118 

R  =  1.0 

5.0 

112.0 

.105 

6.0 

111  .6 

.091 

IV 

1.007 

73.6 

.0 

Old  cast-iron  pipe. 

2.32 

75.5 

.023 

m  =  0.45 

5.075 

75.1 

.02 

R  =  0.1995 

6.801 

75.2 

.02 

12  .576 

75.3 

.02 

V 

Heavily     in  crusted     cast-iron     pipe.     Twenty- 
five  years  in  use. 

1.60 
2.70 
3.60 

64.0 
60.0 
59.0 

0.948 
0.900 
0.874 

m  =  0.30 
R  =  0.416 

4.50 

58.0 

0.858 

VI 

Channel  of  dry  rubble  masonry  of  large  stones, 
bed  somewhat  damaged. 
Six  years  old. 

8.442 
8.905 
9.181 
9.427 

57.4 
54.3 
50.7 
49.6 

0.890 
0.842 
0.784 
0.769 

m  =  0.30 

10  .145 

47.2 

0.731 

R  =  0.19 

VII 

0.50 

126.9 

1.089 

Short  conduit. 

1.0 

116.6 

1  .00 

Wrought-iron  riveted  pipe,  somewhat  rusty. 

1.5 

111  .9 

0.959 

2.0 

109.4 

0.938 

2.5 

109.0 

0.934 

Length  152.9  feet. 

3.0 

108.2 

0.928 

Diameter  8.58  feet. 

3.5 

107.0 

0.917 

m  =  0.54 

4.0 

106.2 

0.910 

R  =  2.145 

4.5 

105  .6 

0.905 

VARIATION  OF   THE   COEFFICIENT    C  21 

Variation  of  the  Coefficient  C  with  the  Roughness  of  the  Wet  Perimeter 

of  a  Conduit. 

Although  the  primary  formula  v  =  y  R  Vs  does  not  give 
satisfactory  results  when  applied  to  all  classes  of  conduits  it 
may  be  made  the  basis  of  formulae  of  general  application. 

Regarding  y  Vr  as  an  approximate  value  of  c  expressions 
may  be  found  defining  the  variation  of  c  with  the  roughness  of 
the  wet  perimeter  as  depending  on  Vr.  The  primary  value  of 
c  from  which  its  variations  with  the  slope  or  the  velocity  of  flow 
must  be  derived  is  that  value  which  corresponds  to  a  velocity 
of  one  foot  per  second. 

In  order  to  retain  if  possible  a  straight  line  formula  we  may 
choose  the  expression 

c  =  (y  Vr)  1  +  m, 

m  indicating  the  condition  of  the  wet  perimeter  of  the  conduit. 

For  a  primary  determination  of  y  and  m  Darcy's  values  of  c 
for  clean  iron  pipes  were  selected.  These  data  give  c  =  112.0 
for  R  =  1.0  and  c  =  80.4  for  R  =  0.0208  (or  a  one-inch  pipe). 

These  values  of  c  are  merely  average  values  found  by  Darcy 
from  a  great  number  of  experiments  on  clean  pipes,  which,  how- 
ever, did  not  include  pipes  of  great  diameters.  Taking  50  as  a 
trial  value  for  y  we  find 

112.0 

=  =  2.24,  hence  1  +  m  =  1  +  1.24 


4 

if  =  4.21,  hence  1  +  m  =  1  +  3.21. 


50  Vr 

Dividing  1.24  by  3.21  that  the  quotient  is  0.386.  This  is  almost 
equal  to  0.38,  the  fourth  root  of  0.0208,  the  value  of  R  for  the 
one-inch  pipe.  We  have  consequently  in  both  instances 

4,-  1.24 

c  =  (50  Vr)  1   f  -47=- 

/YY) 

or  in  general          c  =  (50  Vr)  1  +  -57=- 

vr 

Testing  this  formula  by  experimental  data  pertaining  to  flow 
in  conduits  differing  widely  in  their  degree  of  roughness  it  did 


22 


THE   FLOW   OF    WATER 


not  prove  entirely  satisfactory.  As  already  stated,  Darcy's 
experiments  were  made  on  conduits  of  comparatively  small 
diameters  and  his  coefficients  for  the  larger  conduits  do  not  quite 
agree  with  those  found  by  recent  experiments.  For  the  final 
determination  of  the  value  of  y  we  choose  the  graphical  method. 


If 


then 


=  (If  ^r)  1  +  -TF- 

Vr 


m 


y 


This  is  the  equation  of  a  straight  line   (Fig.  1)  having  for 

1  C 

abscissae  values  of  -j—  -  ,  for  ordinates  values  of  —  j—  and  having 

Vr  yvr 


Values  of 


FIG.  1. 

1.0  as  the  common  distance  from  the  axis  of  abscissae  where  all 
the  lines  intersect  the  axis  of  ordinates;  the  tangent 

—  _10 

y  ^Jr~ 


of  the  angle  a  b  c  will  give  the  value  of  m.    Identical  values 
will  be  obtained  by  putting 

7 


selecting  data  in  which  v  =  1.0  foot  per  second. 


VARIATION   OF   THE   COEFFICIENT   C  23 

Experimental  data  giving  values  of  c  corresponding  to  a  veloc- 
ity of  one  foot  per  second  are  not  numerous  while  those  giving 
values  of  c  corresponding  to  a  velocity  of  one  metre  per  second 
are  quite  abundant,  this  coming  nearer  to  being  an  average 
velocity.  On  this  account  data  given  in  metric  measure  were 
chosen,  taken  chiefly  from  the  writings  of  Darcy-Bazin. 

After  numerous  trials,  and  using  all  the  reliable  material 
available,  a  constant  value  of  y  and  corresponding  values  of 
m  were  found,  producing  a  straight  line  in  every  instance.  As 
our  subsequent  work  depends  much  on  the  reliability  of  this 
constant,  great  pains  were  taken  to  find  its  exact  value.  In 
metric  measure  its  value  is  equal  to  50.0  for  which  in  English 
measure  we  substitute  66.0.  We  have  consequently  for  the 
value  of  c  corresponding  to  a  velocity  of  one  foot  per  second 

c  =  66         1  +  -> 


or,  reducing  to  a  straight  line 

c  =  66  (fa  +  m). 

As  66  (t/r  +  m)  =  c  =  y  ?-2 


and                          (66  (^r  +  m))*  =  c2  =  -j- 
we  have  z  = 7^- 


or  z  = 


(66  (Vr  +  m))2 

0.01478 
(Vr  +  m)2' 


As  primary  expressions  for  the  velocity,  in  most  instances 
true  only  when  the  velocity  is  equal  to  one  foot  per  second  we 
have  now  the  formulae 

v  =  66  (-Vr  +  m)  VrTs.  ....         (1) 


24  THE   FLOW   OF   WATER 

In  the  formula  c  =  66  (Vr  +  m),  when  applied  to  calculations 
of  flow  in  channels  in  earth  of  a  great  degree  of  roughness  of 
the  bed,  the  coefficient  m,  which  indicates  the  degree  of  roughness 
will  have  a  negative  value  and  c  will  in  consequence  vanish  for 
very  small  values  of  Vr.  To  avoid  this  defect  the  formula  may 
be  written,  when  applied  to  channels  in  earth,  so  that  it  reads 

66  (Vr  +  1)     _  66  (Vr  +  Vr) 
=         ~ 


Vr 


in  which  K  is  a  coefficient  increasing  in  value  with  increasing 
roughness  of  the  wet  perimeter.  The  relation  between  m  and 
K  is  given  by 

K--M--1.0 

1  +  m 


Variation  of  the  Coefficient  C  with  the  Velocity  of  Flow. 

A. 

The  characteristics  which  distinguish  water  from  a  perfect 
fluid  are  its  adhesive  qualities,  its  viscosity.  All  fluids,  includ- 
ing gases,  have  these  qualities  in  a  greater  or  lesser  degree.  It 
is  even  asserted  that  solids  like  ice  become  viscous  under  great 
pressures.  The  adhesive  qualities  of  tar  or  crude  oil  are  apparent 
to  the  eye,  those  of  other  fluids  can  only  be  inferred  from  their 
effects. 

To  its  viscosity  is  due  the  fact,  that  water  flowing  in  a 
channel  perfectly  smooth,  is  not,  in  accord  with  the  law  of 
falling  bodies,  continually  increasing  in  speed.  The  retarding 
forces  due  to  viscosity  equalize  the  accelerating  forces  due  to 
gravity  and  distance  fallen  through,  the  speed  of  the  water 
shows  no  increase  from  point  to  point,  in  other  words,  the 
motion  is  uniform. 

The  layer  of  water  immediately  in  contact  with  the  walls  of 
the  channel  in  which  it  flows  does  not  change  except  by  diffusion ; 


VARIATION   OF   THE   COEFFICIENT   C  25 

it  is  held  fast  by  surface  adhesion.  If  the  wall  is  perfectly 
smooth  there  is  consequently  no  friction  between  it  and  the 
fluid  directly  in  contact;  the  resistances  to  flow  are  entirely  due 
to  shearing  stresses  between  the  infinitely  fine  film  coating 
the  wall  and  the  moving  body  of  water. 

Frictional  resistances  are  always  proportional  to  the  areas  of  \ 
the  surfaces  in  contact;  surface  areas  near  the  periphery  of  a 
conduit  are  always  greater  than  near  the  centre  and  the  retarda- 
tion will  in  consequence  be  greater  and  the  velocity  less. 

This  decrease  of  speed  from  the  centre  towards  the  periphery 
is  in  a  measure  counteracted  by  difference  of  pressure.  Greater 
velocities  are  always  accompanied  by  a  corresponding  fall  of 
pressure  and  the  pressure  in  the  centre  is  in  consequence  less 
than  near  the  wall.  This  difference  of  pressure  continually 
tends  to  draw  the  water  towards  the  centre  and  thus  to  equalize 
the  speeds.  When  this  equalizing  tendency  is  for  a  moment 
interrupted,  we  suddenly  perceive  a  wave  or  flash-like  motion, 
clearly  indicating  the  speed  the  water  would  acquire  were  it  not 
for  the  resistances  near  the  periphery.  In  conduits  having 
smooth  walls  the  equalization  of  velocities  is  performed  so 
rapidly  that  a  difference  of  speed  between  the  centre  and  the 
periphery  is  scarcely  perceptible.  A  wave-like  rotation  is  set 
up  and  the  water  glides  through  the  conduit  very  much  like  a 
bullet  through  a  rifled  channel. 

Let  R  be  the  force  required  to  keep  up  the  flow  of  a  liquid  in 
two  parallel  planes  past  each  other,  let  the  surface  area  of  each 
plane  be  A,  let  the  respective  distances  of  the  two  planes  from  a 
common  plane  of  reference  be  Dt  and  Z)0,  let  the  velocities  be  vt 
and  vQ  and  e  a  coefficient  indicating  the  degree  of  viscosity  of  the 
liquid  and  we  have: 

„  _    eA  (vl  -  v0) 


or:  the  resistance  is  proportional  to  the  degree  of  viscosity  into 
the  area  and  the  relative  velocity  v1  -  VQ,  the  whole  divided  by 
the  difference  in  the  distance  of  the  two  layers  from  a  common 
plane  of  reference. 


26  THE   FLOW  OF   WATER 

For  a  circular  conduit  the  total  force  required  to  set  up  motion 
in  a  stream  line  is  given  by 

4el  (v,  -  v0)         2flv 

£1  =  -  ;  -  ;  —     ~r    - 

Tj2  -  r02  rt 

in  which  rl  is  the  semi-diameter  of  the  conduit,  r0  a  distance  from 
the  axis  of  the  conduit,  I  its  length,  and  /  the  coefficient  indicating 
the  degree  of  roughness  of  the  surface. 

This  indicates,  that  the  resistance  due  to  viscosity  is  least  in 
the  centre  or  the  axis  of  the  conduit  where  r02  =  0  and  greatest 
at  the  periphery  where  rt2  —  r02  =  0. 

The  last  expression  gives  for  the  velocity  of  flow  in  a  circular 
conduit 


/ 

v  =      / 


0  , 
2gh  + 


from  which,  the  coefficients  e  and  /  being  known,  the  value  of  v 
and  the  discharge  may  be  computed.  The  coefficient  e  depends 
for  its  value  on  the  temperature  of  the  liquid ;  its  value  diminishes 
rapidly  with  increase  of  temperature  and  is  five  times  less  for 
water  at  the  boiling  point  than  for  water  at  the  freezing  point. 
According  to  Mayer  its  values  are  (in  c.  g.  s.  units) : 

at  0.6°  Celsius  e  =  0.0173 
at  10°  Celsius  e  =  0.0131 
at  20°  Celsius  e  =  0.010 
at  45°  Celsius  e  =  0.005833 
at  90°  Celsius  e  =  0.00339. 

The  influence  of  the  temperature  on  the  flow  of  water  through 
capillary  tubes  has  been  minutely  studied  by  Poisseule. 
Slichter  has  demonstrated  the  immense  influence  of  the 
temperature  on  the  movement  of  water  through  permeable 
strata  and  Saph  and  Shoder  have  shown  its  influence  on  the  dis- 
charge of  pipes.  A  tube  having  a  diameter  of  0.5  millimetre 
(0.02  inch)  or  less  is  considered  to  be  a  capillary  tube. 


VARIATION   OF  THE   COEFFICIENT  C 


27 


Poisseule's  experiments  demonstrated,  that   the  velocity   of 
flow  in  such  tubes  is  equal  to 


and  the  discharge  to 


This  shows,  that  in  capillary  tubes  the  velocity  is  proportional 
to  the  head  and  not  to  the  square  root  of  the  head,  to  the  square 
of  the  radius  and  not  to  its  square  root. 

In  investigations  on  the  movement  of  water  through  porous 
strata  it  has  been  found,  that  the  velocity  of  flow  is  proportional 
to  the  square  of  the  diameter  of  the  soil  grains  through  which 
the  water  percolates;  from  which  it  follows  that  it  is  also  pro- 
portional to  the  square  of  the  voids  between  the  soil  grains. 
The  general  equation  for  the  movement  of  water  in  a  per- 
meable stratum  may  be  written  (v  and  Q  per  minute) 

-  0.0189d2  (0.7  4-  0.03  t'c) 


2L 

Q  =  mv  hb. 

In  these  equations  h  is  the  elevation  of  the  water  table  at  the 
point  of  efflux,  h  +  z  its  elevation  at  the  distance  L,  d  the 


FIG.  2. 


diameter  of  the  soil  grains  in  millimetres,  t°  0,  the  temperature  of 
the  water  in  degrees  centigrade,  m  the  percentage  of  the  voids  in 
the  material,  b  the  breadth  of  the  stratum. 


28 


THE  FLOW  OF   WATER 


The  elevation  of  the  water  table  at  the  distance  x  from  the 
point  of  efflux  is  equal  to 


y 


2Qx 


hm  0.0198  d2  b 
The  discharge  of  a  well  is  given  by 


Q 


T 
log  L  -  log  R 


0.0189  d*  (0.7  +  0.03 


in  which  R  is  the  semi-diameter  of  the  well,  and  the  logarithms 
the  Naperian 


FIG.  3. 

The  elevation  of  the  water  table  at  a  distance  x  from  the  well  is 
given  by 


The  surface  is  consequently  a  logarithmic  curve.  These 
equations  serve  to  illustrate  that  between  the  movement  in 
capillary  tubes  and  in  a  porous  stratum  there  is  only  this  dif- 

Ji2 

ference,  that  h  is  displaced  by  — - ,  the  velocity  is  not  proportional 

£ 

to  the  head  but  to  the  square  of  the  head. 

On  account  of  internal  motions  the  phenomenon  of  flow  in 


VARIATION    OF   THE    COEFFICIENT  C  29 

pipes  and  other  channels  is  much  more  complex  than  in  capillary 
tubes  or  porous  strata. 

The  equation  for  the  velocity  of  flow  in  a  cylindrical  conduit 
we  have  given  above  may  be  transformed  so  it  will  read 

-  -  %» +  r-JM!T7-v M— 

f  2    ~,  2     I     O      7. 

'1  'n  l^~7't« 


which  shows  that  one  of  the  terms  above  the  line,  denoting  the 
internal  resistance,  is  directly  proportional  to  the  velocity,  the 

p 

other  to  its  square.     This  is  also  true  of  the  two  terms  2-^ 

denoting  the  friction  of  the  fluid  against  the  walls  of  the  conduit. 
Moreover,  /,  the  coefficient  denoting  the  surface  friction,  depends 
for  its  value  on  e,  the  coefficient  denoting  the  internal  friction; 
its  value  is  consequently  modified  by  temperature.  Even  in 
conduits  having  the  smoothest  walls  there  are  always  rotary  and 
wave-like  motions  tending  to  equalize  pressures  to  speeds. 
Wherever  there  are  cross-currents  there  naturally  is  impact,  one 
stream  impinging  on  the  other.  To  this  impact  and  the  attend- 
ing shearing  stresses  between  a  streamline  and  its  surroundings 
are  due  the  increasing  powers  of  the  velocity  to  which  the 
resistances  are  proportional.  Furthermore,  if  the  walls  of  the 
conduit  are  not  perfectly  smooth  there  are  streamlines  constantly 
impinging  on  projections,  however  small  they  may  be. 

Conditions  existing  at  the  entrance,  curves,  elbows,  changes 
of  section,  etc.,  also  affect  the  power  of  the  speed  to  which  the 
total  resistance  is  proportional.  It  was  formerly  assumed  that 
the  resistances  due  to  these  impediments  were  proportional  to 
the  square  of  the  velocity. 

From  experiments  made  by  Hubbel  and  Fenkell  (Detroit)  to 
determine  the  resistances  due  to  curves,  the  writer  found, 
neglecting  curves  the  radius  R  of  which  is  less  than  2.5  diameters 
of  the  conduit,  that  the  resistance  of  a  curve  is  equal  to 


30  THE   FLOW   OF   WATER 

times  the  resistance  of  a  tangent  of  equal  length,  and  the  excess 
of  frictional  resistance  in  a  curve  equal  to 


times  the  resistance  in  a  tangent  of  equal  length.  The  length 
of  tangent  equal  in  frictional  resistance  to  the  resistance  in  a 
curve  of  90°  is  equal  to 


This  evidently  vanishes  when  —  =  (4.9)^  d  =  539.3  and  is  a 

a 

maximum  when  —  =  (4.9)  *  d. 
d 

Hubbel  and  Fenkell's  experiments  were  made  on  30",  16"  and 
12"  conduits,  and  comparison  showed  that  the  influence  of  the 
diameter  on  the  resistance  depends  on  the  value  of  d™.  The 
value  of  z,  or  /,  the  coefficient  of  friction  is  therefore  for  any 
curve. 


.  -  1.0 

360 

in  which  n  is  equal  to  the  number  of  degrees  in  the  curve  and 
d°'45  substituted  for  d™  for  diameter  less  than  a  foot. 

Hubbel  and  Fenkell's  experiments  were  supplemented  by  those 
of  Saph  and  Schoder  on  2-inch  brass  tubes  and  more  recently 
by  those  of  Alexander  on  a  IJ-inch  wooden  tube.  Although  we 
cannot  accept  the  formulae  the  latter  deduced  from  his  own  ex- 
periments and  those  of  Hubbel  and  Fenkell,  Saph  and  Schoder, 
his  experiments  are  valuable  in  indicating  the  powers  of  the 
velocity  to  which  resistances  in  curves  are  proportional.  While 
Alexander's  experiments  show  that  resistances  in  a  curve  are 
proportional  to  the  same  powers  of  the  speed  as  resistances  in  a 
tangent,  provided  there  is  no  shock,  the  experiments  of  Saph 
and  Schoder  indicate  that  the  power  of  the  speed  increases 

rapidly  with  increasing  values  of  —  •    Their  data  indicate  that 

ti 


VARIATION   OF  THE   COEFFICIENT  C  31 

7?  7? 

for  —  =  10  the  resistance  is  proportional  to  V1'8,  for  —  =4  to 
a  a 

72'87.  How  far  this  holds  good  for  diameters  greater  than 
2  inches  we  are  not  prepared  to  say.  It  is  probable,  however, 
that  with  increasing  diameter  the  force  of  the  shock  decreases 
and  the  powers  of  v  with  it. 

It  is  probable  that  the  resistances  due  to  right-angled  entrances 
right-angled  elbows  are  also  proportional  to  powers  of  v  higher 
than  2.0. 

The  effect  of  the  temperature  on  the  variation  of  the  power 
of  v  has  so  far  not  been  determined  with  precision.  Saph 
and  Schoder,  experimenting  with  a  2-inch  brass  pipe,  found  for 
a  rise  of  10°  F.  an  increase  in  the  discharge  of  4%. 

That  the  resistances  to  flow  are  not  proportional  to  the  square 
of  the  speed  was  recognized  long  before  Darcy  and  Bazin 
demonstrated  the  great  influence  of  the  degree  of  roughness  of  the 
walls  of  a  channel  on  its  discharge. 

The  laws  of  fluid  friction  were  first  investigated  by  Coulomb. 
He  states,  that  the  total  resistance  to  motion  is  a  compound  of 
two  factors,  one  being  proportional  to  v,  the  other  to  v2.  Dubois's 
experiments  on  flow  confirmed  this  view  and  from  his  data 
Prony  found  for  the  resistance  the  expression  (in  metric  measure) 

Rv  =  0.000044  v  +  0.000309  v2, 
this  corresponds  to 

ff-?TT' 

Weisbach  put  Prony's  formula  into  the  form 
H  =  0.00741 


v     I  r 
which  in  our  day  is  still  used. 

The  relation  of  the  power  of  the  velocity,  to  which  the  resist- 
ance is  proportional,  to  the  variation  of  the  coefficient  c  with  the 
velocity  is  such,  that  c  remains  constant  for  all  velocities  if  the 
resistance  is  proportional  to  v2;  it  increases  with  increase  of 
velocity  if  the  resistance  is  proportional  to  v*~*,  and  decreases  if 
the  resistance  is  proportional  to  v**x  • 


32  THE  FLOW  OF  WATEK 

B. 

If  the  value  of  the  coefficient  c  corresponding  to  any  velocity 
is  divided  by  its  value  corresponding  to  a  velocity  of  one  foot  per 
second;  the  quotient  is  a  variable  which  we  will  call  the  coefficient 
of  variation  of  c  and  denote  by  (a).  Hence 

c 

a 


66  (Vr  +  m) 
While  the  term  i 


[66  (Vr  +  m)]2 

represents  the  frictional  resistance  per  unit  area  of  surface  at 
unit  velocity,  the  term 

c 

66  (t/r  +  m) 

indicates  the  power  of  the  velocity  to  which  the  resistance  is 
proportional.  We  shall  presently  see,  that  under  normal  con- 
ditions, that  is  if  resistances  proportional  to  different  powers  of 
v  do  not  enter,  the  coefficient  a  is  merely  a  root  of  v. 

An  analysis  of  the  values  of  a  found  in  Column  4,  Table  II, 
shows  that  its  value  does  not  entirely  depend  on  the  velocity, 
but  is  affected  by  the  degree  of  roughness  of  the  walls  of  the 
conduit,  by  its  length  and  alignment,  by  conditions  existing  at 
the  entrance,  by  changes  of  section,  etc.  According  to  the 
manner  in  which  the  coefficient  a  is  affected  we  may  classify 
conduits  as  follows: 

1.  Long  straight  conduits  without  internal  obstructions  and 
a  great  degree  of  smoothness  of  the  wet  perimeter. 

2.  Long  conduits  of  a  great  degree  of  smoothness  of  the  wet 
perimeter  but  with  some  easy  curves  or  other  impediments,  also 
long  straight  conduits  of  a  fair  degree  of  smoothness  of  the  wet 
perimeter. 

3.  Long  conduits  of  a  great  or  fair  degree  of  smoothness  of 
the  wet  perimeter  but  with  sharp  curves,  angles  or  other  impedi- 
ments to  flow. 

4.  Conduits  whose  walls  are  coated  with  rust,  slimy  or  sticky 
substances. 


VARIATION   OF  THE   COEFFICIENT  C  33 

5.  Conduits  of  a  great  degree  of  roughness  of  the  wet  peri- 
meter;  badly  tuberculated  pipes,  damaged  masonry,  channels 
in  earth  with  sharp  bends,  bars  or  other  obstructions. 

6.  Short  conduits. 

For  classes  1  and  2  the  resistances  are  proportional  to 
powers  of  v  less  than  2.0  and  the  coefficients  c  and  a  continue 
to  increase  in  value  with  increasing  velocity.  For  the  third 
class  some  resistances  proportional  to  a  power  higher  than 
2.0  enter,  a  increases  with  increase  of  velocity  and  then 
decreases.  For  class  4  the  resistance  is  proportional  to  v2 
or  nearly  so  and  a  is  constant.  For  classes  5  and  6  the  resist- 
ances are  proportional  to  powers  of  v  higher  than  2.0  and  a 
continually  decreases  with  increasing  velocity. 

C. 

We  have  so  far  only  found  expressions  for  the  value  of  c 
corresponding  to  a  velocity  of  one  foot  per  second.  These  give 
for  the  velocity 

v  =  66    (t/r  +  m)  Vrs (1) 


2    H  ....     (2) 


0.01478     L 
(Sir  +  my  R 

We  will  now  proceed  to  find  in  what  relation  the  'value  of  v 
as  found  from  the  formula 

v  =  66  (tfr  +  m)  Vr  .  s 

stands  to  the  true  mean  velocity  in  all  cases  where  v  is  more 
or  less  than  unity  or  the  value  of  a,  the  coefficient  of  variation, 
is  affected  by  the  conditions  we  have  enumerated.  Using  the 
exponential  equation 

(66  (^r  +  m) 


_[ 
L 


(66  (r  +  m) 
which  gives 

log  yt  _  —  _  log  v0 


x  = 


log    (66  (#r  +  m)  Vr.s),  -  log    (66  (Vf  +  m)  Vrs), 


OF   THE 

UNIVFRRITY 


34  THE  FLOW  OF   WATER 

we  find  from  the  experimental  data  given  by  Darcy  and  Hamilton 
Smith  for  straight  or  nearly  straight  clean  cast-iron,  wrought- 
iron  and  sheet-iron  riveted  pipes  of  all  diameters  and  for  velocities 
up  to  20  feet  per  second 

*-|i 

in  other  words,  from  the  data  given  by  Darcy  and  Hamilton 
Smith  we  find,  that  the  true  mean  velocity  is  equal  to 

v  =  (66  (vV  +  m)^r.s)*    .....    (3> 
which  may  be  written 


v  =  66  (-r  +  m)      ree  (^r  +  m) 


hence  the  coefficient  a,  indicating  the  variation  of  the  coefficient 
c  with  the  velocity  is  equal  to 

a  =  466  (\V  +  m)  \/r~7s. 


From  Formula  3  we  have  also 

yt  =  (66  tJr  +  m) 

consequently 

a  =  V* 

and  y*  =  iV(66  A/r  +  m)  VrTsT 

Table  III  contains  a  number  of  experimental  data  relating  to 
flow  in  conduits  under  pressure.  They  are  purposely  selected 
in  order  to  show  the  variation  of  the  coefficient  c  as  affected  by 
various  conditions  of  flow. 

The  values  of  the  coefficient  a  found  in  columns  3,  6  and  9 
show  that  for  1-inch  pipes  of  tin  and  wrought  iron,  for  sheet-iron 
riveted  pipes  up  to  2.43  feet  in  diameter,  for  new  cast-iron  pipes 
up  to  1.393  feet  in  diameter,  for  pipes  of  planed  shares  up  to  4.5 
feet  in  diameter  the  coefficient  a  is  equal  to  y^  or  nearly  so. 
The  fact  that  a  =  F  holds  good  for  a  tin  or  wrought-iron  pipe 
1  inch  in  diameter,  and  also  for  a  pipe  of  planed  staves  54  inches 
in  diameter  allows  us  to  conclude,  that  it  holds  good  also,  be- 
tween these  limits  for  other  conduits  having  walls  of  a  similar 
degree  of  roughness  such  as  asphalt-coated,  cast  and  wrought- 
iron  or  cement-lined  pipes. 


VARIATION  OF  THE   COEFFICIENT  C  35 

This,  however,  holds  good  only  when  the  value  of  —  ,  the  ratio 

ct 

between  the  length  of  a  pipe  and  its  diameter,  is  at  least  1,000. 

For  lesser  values  of  —  the  value  of  (a)  decreases  with  —  -  •    The 
a  a 

experimental  values  given  by  Stearns  and  Fitzgerald  relating  to 
flow  in  four-foot  cast-iron  pipes  indicate  this  plainly.     In  the 

case  of  the  four-foot  Sudbury  conduit  (Stearns)  the  ratio  —  is 
equal  to  439. 

If  the    formula  v  =  (66  (tfr  +  m)  Vr  .  s)*   is  put  into  the 
form 


V  = 


the  term 


-Vr  +  m)2  R 


includes  all  the  resistances,  those  due  to  the  velocity  itself, 
those  due  to  the  entrance,  and  those  due  to  the  friction  of  the 
fluid  against  the  walls  of  the  conduit. 

The  loss  of  head  due  to  the  velocity  itself  is  proportional  to 
the  square  of  the  speed;  resistances  due  to  the  entrance  are  pro- 
portional, according  to  the  nature  of  the  entrance  all  the  way 
from  the  square  of  the  speed  up  to  its  cube.  An  average  value  is 
probably  2.5. 

The  value  of  the  coefficient  of  resistance  due  to  the  velocity 
itself  is  equal  to  1.0;  the  value  of  z0,  the  coefficient  representing 
the  resistance  due  to  the  entrance  is,  according  to  Weisbach,  for 
a  well  rounded  entrance,  equal  to  0.505;  hence  the  value  of  z1} 
the  coefficient  representing  the  resistance  due  to  the  walls  of 
the  conduit,  is  equal  to 

°'01478    L    -  1.505. 


+  m)2  R 


36  THE   FLOW   OF   WATER 

If  the  conduit  is  long,  above  1,000  diameters  in  length,  1.505 
is  a  quantity  small  in  comparison  with 

0.01478     L 


Vr  +  m)2   R 

and  does  in  consequence  not  appreciably  affect  the  variation  of 
c.  With  decreasing  length  of  the  conduit,  however,  the  ratio 
between  the  two  quantities  changes  at  an  increasing  rate  and 
more  and  more  affects  the  variation  of  c.  In  the  case  of  the 
four  foot  Sudbury  conduit  (Stearns),  we  have  the  following 
data,  taking  m  =  0.97 : 

v  =  3.738,  H  =  1.2421  ft.,  L  1747  ft. 

0.01478     L 

—  =  6.656 


(1  +  0.97)2  R 
6.656  -  1.505  =  5.151  =  zr 
(3jJ38)2   X  1.0  =  0.217  =  h  =  loss  of  head  due  to  velocity. 

«7 

(3'738)2>5  X  0.505  =  0.2119  =  h0  =  loss  of  head  due  to  entrance. 

0.217  +  0.2119  =  0.4289  =  h  +  h0. 

1.2421  -  0.4289  =  0.8182  =  ^  =  loss  of  head  due  to  friction  in 
the  pipe  itself. 

Using  the  formula  -£—  =  vx  and  inserting  values  we  have 
64.4  X  0.8142  =  = 


5.151 

Dividing  log  10.17  =  1.0073209  by  log  3.738  =  0.5736293 
the  quotient  =  1.76  very  near. 

Consequently  the  frictional  resistance  in  the  pipe  itself  is 
proportional  to  v1'76,  corresponding  closely  to  FT",  the  value  we 
have  found  for  long  pipes. 

The  data  relating  to  flow  in  a  riveted  flume  8.58  feet  in  diameter 
and  152.9  feet  long  (Herschel,  Holyoke  Testing  Flume)  show  the 
great  influence  of  the  length  of  the  conduit  on  the  variation  of  c 
most  plainly. 


VARIATION   OF   THE   COEFFICIENT  C 


37 


TABLE  III. 

EXPERIMENTAL  DATA  SHOWING  EXTENT  OF  VARIATION  OF  c  WITH  THE 
VELOCITY  OF  FLOW. 


Tin  pipe,  straight. 
Dubuat. 

Wrought-iron  pipe. 
Darcy. 

Asphalt-  coated  riveted 
pipe.     Darcy. 

d  =  0.0888  ft. 

d  =  0.1296  ft. 

d  =  0.271  ft. 

Z/not  given. 

L  =  372  ft. 

L  =  365  ft. 

m  =  0.98 

m  =  0.83 

m  =  0.94 

a   =F* 

a  =  H 

a   =  V» 

V 

c 

a 

V 

c 

a 

V 

c 

a 

0.141 

67.6 

0.751 

0.205 

76.9 

0.932 

0.328 

76.7 

0.80 

0.772 

82.8 

0.92 

0.858 

82.3 

0.995 

1.171 

99.9 

1  .043 

1.183 

91.4 

1.019 

2.585 

92.9 

1  .112 

3.117 

108.4 

1.132 

2.546 

98.9 

1.099 

6.3 

99.8 

1.21 

6.148 

117.4 

1  .223 

2.606 

100.4 

1.115 

8.521 

100.0 

1.212 

10  .535 

124.0 

1.274 

5.223 

111.4 

1  .237 

12  .786 

124.3 

1.298 

Asphalt-coated  riveted 
pipe.     Darcy. 

Asphalt-coated  riveted 
pipe.     H.  Smith. 

Asphalt-coated  riveted 
pipe.     H.  Smith. 

d  =  0.643  ft. 

d  =  0.911  ft. 

d  =  1.229  ft. 

L  =  365  ft. 

L  =  700  ft. 

L  =  700  ft. 

m  =  0.92 

m  =  0.68 

m  =  0.69 

a   =  V? 

a   =  7? 

a   =  7^ 

V 

c 

a 

V 

c 

a 

V 

c 

a 

0.591 

104.1 

1.013 

4.712 

107.1 

1.19 

4.283 

111  .6 

1.181 

1.529 

106.2 

1  .035 

6.094 

110.6 

1  .229 

6.841 

117.8 

1  .246 

1.53 

115.6 

1  .125 

6.927 

111  .5 

1  .24 

7.314 

119.1 

1  .261 

5.509 

125.4 

1.22 

8.659 

113.4 

1  .26 

8.462 

119.1 

1.26 

9.0 

130.2 

1  .267 

10  .021 

115.5 

1.283 

10  .593 

121.6 

1  .285 

19.72 

141.0 

1.372 

12.09 

121.3 

1.280 

38 


THE   ELOW   OF   WATER 


TABLE  III.  —  Continued. 


Asphalt-coated  cast- 

Asphalt-coated   cast- 

Asphalt-coated  cast- 

iron  pipe.     Darcy. 

iron  pipe.     Hubbel 
&  Fenkell. 

iron  pipe.     Lampe. 

d   =  0.4495  ft. 

d   =  1.0ft. 

d  =  1.373  ft. 

L  =  366  ft. 

L  not  given. 

L  =  26,000  ft. 

m  =  0.90 

m  =  0.83 

m  =  0.83 

a   -F* 

a   =  V? 

a   =  V? 

V 

c 

a 

V 

c 

a 

V 

c 

a 

0.489 

94.1 

0.96 

1  .0 

101  .5 

.0 

1  .577 

110.5 

1.072 

2  .503 

108.4 

1.107 

2.0 

109.6 

.06 

2.489 

114.1 

1.107 

5.625 

112.5 

1.15 

3.0 

114.6 

.13 

2.709 

114.6 

1.112 

11  .942 

113.5 

1.16 

4.0 

118.3 

.166 

3.090 

119.4 

1.162 

15  .397 

112.2 

1.15 

5.0 

121  .3 

.196 

Redwood  Stave  Pipes. 
A.  L.  Adams. 

Cedar  Stave  Pipe. 
Th.  A.  Noble 

Cedar  Stave  Pipe. 
Th.  A.  Noble. 

d  =  1.166  ft. 

d   ==  3.667  ft. 

d   =  4.5  ft. 

L  =  80,006  ft. 

Lnot  given. 

Lnot  given. 

m  =  0.93 

m  =  0.50 

m  =  58 

a   =  V? 

a   =  V* 

a   =  V^ 

V 

c 

a 

V 

c 

a 

V 

c 

a 

0.698 

97 

0.908 

3.468 

110.1 

.134 

2.282 

116.8 

1  .095 

0.698 

101 

0.926 

3.522 

108.6 

.12 

2.276 

115.5 

1  .086 

0.751 

104 

0.953 

3  .685 

110.9 

.144 

2.650 

119.9 

1  .12 

0.691 

105 

0.963 

3  .853 

112.6 

.163 

3  067 

122.1 

1  .151 

1.167 

109 

1.0 

3.964 

112.9 

.164 

3  .045 

121  .4 

1  .138 

1  .531 

112 

1  .027 

3  .972 

113.1 

.164 

3.408 

123.7 

1  .164 

1.181 

113 

1  .036 

4.415 

113.7 

.164 

3.724 

125.2 

1.176 

4.635 

114.9 

.183 

3.929 

126.2 

1.179 

4.831 

115.5 

.190 

4.688 

129.0 

1.205 

VARIATION  OF  THE   COEFFICIENT  C 


39 


TABLE  III.  —  Continued. 


New  steel-riveted  pipe. 

New  steel-riveted  pipe. 

New  steel-riveted  pipe. 

Herschel. 

Herschel. 

Marx-Wing. 

d   =  3.5  ft. 

d   =  4.0  ft. 

d  =  6.0  ft. 

L  =  81,339  ft. 

L  =  24,648  ft. 

L  =  4,367  ft. 

m  =  0.56 

m  =  0.47 

m  =  0.50 

a  =  V& 

a   =  V& 

a   =  V^ 

V 

c 

a 

V 

c 

a 

V 

c 

a 

1.0 

101.0 

1.0 

1 

97.1 

1.0 

1.07 

103.5 

1.0 

2.0 

104.3 

1.032 

2 

101.3 

1  .043 

1.67 

108.0 

1  .024 

3.0 

106.4 

1  .053 

3 

102.2 

1  .052 

2.14 

113.0 

1  .091 

4.0 

107.8 

1  .067 

4 

104.2 

1.073 

2.50 

108.0 

1  .024 

5.0 

108.4 

1.073 

5 

105.1 

1.083 

3.0 

112.0 

1.082 

6.0 

108.5 

1.074 

6 

105.2 

1.084 

3.84 

113.0 

1.091 

Asphalt-coated  cast- 

Cleaned  cast-iron  pipe. 

Cedar  stave  pipe 

iron  pipe.     Stearns. 

Fitzgerald. 

Marx-Wing. 

d  =  4.0  ft. 

d  =  4.0  ft. 

d  =  6.0  ft. 

L  =  1,747  ft. 

L  not  given. 

L  =  4,000  ft. 

m  =  0.97 

m  =  0.98 

m  =  0.66 

a,  =  FIT 

a  =  yr§ 

a   =  V&i 

V 

c 

a 

V 

c 

a 

V 

c 

a 

3.738 

140.1 

1.077 

2.472 

137.5 

1.051 

1.0 

116.0 

1.0 

4.965 

142.1 

1  .093 

3.723 

139.1 

1.064 

1.5 

118.7 

1.023 

6.193 

144.1 

1.109 

4.796 

141  .1 

1.085 

2.0 

119.9 

1.032 

6.141 

143.6 

1.100 

3.0 

121.4 

1.046 

4.0 

122.0 

1.051 

5.0 

122.4 

1  .055 

6.0 

122.5 

1  .056 

40  THE   FLOW   OF   WATER 

The  experimental  data  relating  to  flow  in  riveted  conduits 
show  great  diversities  both  in  the  values  of  m  and  a. 

The  coefficient  m  is  equal  to  0.94  for  a  riveted  pipe  0.270  feet 
in  diameter  (Darcy)  and  equal  to  0.51  for  a  butt-jointed  riveted 
pipe  six  feet  in  diameter  (Marx  -Wing).  This  great  difference 
in  the  values  of  m  is  mainly  due  to  the  size  of  the  rivet  heads. 
In  pipes  of  small  diameters  the  rivet  heads,  especially  when 
coated  with  asphalt,  do  not  offer  an  appreciable  impediment  to 
flow.  In  large  conduits,  however,  their  size  is  such,  that  they 
not  only  produce  constriction  of  the  section,  but  also  vortex 
motions,  thus  reducing  the  discharge  in  a  twofold  manner. 
From  data  relating  to  flow  in  steel-riveted  pipes  exceeding  three 
feet  in  diameter,  given  by  Herschel,  and  by  him  considered  the 
most  reliable  (see  "  Herschel  "115  Experiments),  we  find  that 
the  coefficient  of  variation  of  c  for  these  conduits  is  fairly,  though 
not  precisely,  equal  to 


(Vr  +  m)  Vf~7. 
Consequently 


(66  (Vr  Hh  m)  Vr  .  s)i?  . 

.     .    .          (5) 

f         2gH          1* 

(6"\ 

0.01478      L 

.(^r  +  m)2    ^J 

or  v  = 


This  value  of  a  =  V  we  find  to  hold  good  also  for  flow  in 
rectangular  pipes  (Darcy). 

The  experimental  data  relating  to  flow  in  old  iron  pipes,  those 
not  heavily  incrusted  or  tuberculated  show  that  the  coefficient  c 
does  not  vary  to  any  extent  with  variations  in  the  value  of  v. 
For  this  class  of  conduits  we  have  consequently 

a  =  1.0. 

The  data  relating  to  flow  in  badly  incrusted  or  heavily  tuber- 
culated pipes  indicate  a  decrease  in  the  value  of  c  with  increasing 
velocity.  Using  as  before  the  equation 

log  v,  log^p 


x  = 


log   (66    (Vr  +  m)  Vrs),  -  (66  (Vr  +  m)  Vrs)( 


VARIATION   OF  THE  COEFFICIENT  C  41 


we  find  for  incrusted  pipes: 
x  =  if,  hence  a  =    — 


19/  _ 

V™      V66  (^r  +  m)  Vr.s 

and  for  very  badly  tuberculated  pipes: 
x  =  y90,  hence  a  =  -  -  = 


^66  (>/r  +  m)  Vr  .  a 

The  experimental  data  relating  to  flow  in  a  12-foot  brick 
sewer  at  Milwaukee,  a  7.5-foot  brick  sewer  at  Dorchester  Bay,  in 
a  siphon  aqueduct  of  119  feet  cross-section  at  the  river  Elvo  all 
show  a  slight  decrease  in  the  value  of  c  with  increasing  velocity. 

This  decrease  is  due,  in  the  first  two  cases,  partly  to  the  fact 
that  these  conduits  are  discharging  under  water  against  a 
hydraulic  counter  pressure,  partly  it  is  due  to  the  greater  viscosity 
of  the  sewerage  and  partly  also  to  the  relative  shortness  of  these 
conduits.  In  the  case  of  the  siphon  aqueduct,  its  length  is  so 
short  comparatively,  that  it  can  only  be  considered  as  a  short 
pipe,  conditions  being  much  the  same  as  in  the  case  of  the 
Holyoke  Testing  Flume. 

D. 

The  variation  of  the  coefficient  c  as  deduced  from  experi- 
mental data  relating  to  flow  in  conduits  under  pressure  may  be 
summarized  as  follows: 

1.  For  long,  straight  conduits  fairly  clean,  such  as  pipes  of 
glass,  tin,  lead,  galvanized  iron,  cast  and  wrought  iron,  planed 
staves,  cement,  riveted  pipes  up  to  3  feet  in  diameter,  the  coeffi- 
cient of  variation  of  c  is  equal  to 

a  =  7* 

and  the  frictional  resistance  is  proportional  to  7"*". 

2.  For  pipes  rectangular  in  section,  for  riveted  pipes  exceeding 
3  feet  in  diameter,  for  those  enumerated  under  (1)  between  300 
and  1,000  diameters  in  length,  the  coefficient  of  variation  of  c 
is  equal  to 

a  =  7fs 

and  the  frictional  resistance  is  proportional  to  7~*~. 


42  THE   FLOW  OF   WATER 

3.  For  the  classes  of  pipes  enumerated  under  (1)  and   (2) 
discharging  against  a  hydraulic  counterpressure,   or  between 
100  and  300  diameters  in  length,  for  old  pipes  not  incrusted  or 
tuberculated  the  coefficient  c  does  not  appreciably  vary  with  the 
velocity,  and  consequently 

a  =  1.0 

and  the  frictional  resistance  is  proportional  to  F2. 

4.  For  incrusted  pipes  and  those  enumerated  under  (1)  and 
(2)  less  than  100  diameters  in  length  the  coefficient  of  variation 
of  c  is  equal  to 

1 
a  = 


and  the  frictional  resistance  is  proportional  to 

3.  For  very  heavily  tuberculated  pipes  the  coefficient  of  vari- 
ation of  c  is  equal  to 

1 

a  =  —  i 

v* 

and  the  frictional  resistance  is  proportional  to  Vs  . 

In  our  collection  of  experimental  data  we  find  many  instances 
relating  to  flow  in  one  and  the  same  conduit  which  do  not  fit 
any  of  the  values  of  (a)  enumerated  above  and  which  indicate  : 

1.  First  an  increase  in  the  value  of  c  with  increasing  velocity 
up  to  a  certain  critical  velocity. 

2.  Then  a  decrease  in  the  value  of  c  with  increasing  velocity. 
As  instances  of  this  kind  we  mention: 

Two  new  steel  riveted  pipes  at  East  Jersey,  3.5  and  4  feet  in 
diameter  (Herschel). 

A  cement  lined  pipe  with  elbows  (Fanning). 

This  peculiar  variation  of  c  indicates  the  presence  of  resistances 
which  are  proportional  to  powers  of  the  velocity  greater  than 
2.0,  that  is  resistances  which  produce  shocks.  In  case  of  the 
steel  pipes  the  shocks  are  no  doubt  due  to  the  rivet  heads,  in 
the  second  to  the  elbows  in  the  line  of  the  conduit.  This 
peculiar  variation  of  the  coefficient  c  is  also  very  plainly  indicated 
in  the  data  relating  to  flow  in  channels  of  rough  boards  with 
cleats  nailed  crosswise  to  bottom  and  sides  of  the  channel 


OPEN   CONDUITS  43 

(Darcy-Bazin,  series  12-17).  These  cleats  or  laths  were  1  centi- 
metre thick  and  2.5  centimetres  wide.  In  one  channel  they 
were  spaced  apart  1  centimetre,  in  the  other  5.0.  The  data 
relating  to  flow  in  the  channel  with  the  cleats  spaced  1  centi- 
metre indicate  the  highest  values  of  both  the  coefficients  m 
and  a,  plainly  showing  the  effect  of  the  shock  due  to  the  wider 
spacing  of  the  cleats.  In  the  first  case  the  coefficients  are  m  = 

0.41,  a  =  V™,  in  the  second  m  =  0.03  a  =  — ,  indicating   that 

the  frictional  resistance  was  proportional  in  the  first  case  to 
71'94,  in  the  second  to  F2'25. 


Open  Conduits. 
E. 

An  analysis  of  experimental  data  relating  to  flow  in  open 
conduits  of  permanent  cross-section,  such  as  aqueducts,  flumes, 
etc.,  indicates,  that  the  coefficient  c  is  affected  in  its  variation 
with  the  velocity  by  the  shape  of  the  cross-section,  or  by  the 
depth  of  the  water  in  the  channel. 

For  semicircular  or  well  rounded  channels,  for  the  semi-square 
when  flowing  full,  for  all  sections  for  which  the  mean  hydraulic 
radius  is  equal  to  half  the  depth,  for  the  triangle  with  sides 
inclined  45°  the  variation  of  the  coefficient  c  with  the  velocity 
does  not  seem  to  be  affected  by  slight  variations  in  the  value 
of  r.  For  rectangular  channels,  however,  and  others  having  very 
steep  side  walls  (excluding  those  mentioned  above)  the  varia- 
tion of  c  is  affected  by  the  depth  of  water  in  the  conduit. 

The  coefficient  a  seems  to  have  its  normal  value  in  all  instances 
when  the  depth  of  water  is  equal  to  one-half  the  mean  width  of  the 
channel,  it  increases  in  value  as  the  depth  of  water  decreases, 
and  decreases  in  value  as  the  depth  of  water  increases. 

This  peculiar  influence  of  the  steepness  of  the  walls  of  a 
conduit  on  the  frictional  resistance  has  been  revealed  by  nu- 
merous current  metre  observations  in  rectangular  flumes  and 
aqueducts  and  other  channels  with  steep  side-walls.  It  has  been 


44  THE   FLOW   OF   WATER 

found  that  in  such  channels  the  position  of  the  thread  of  max- 
imum velocity  is  situated  at  a  greater  distance  from  the  surface 
than  in  channels  having  side  walls  more  inclined:  thus  clearly 
indicating  the  retarding  influence  of  the  steepness  of  the  walls. 
Experimental  data  relating  to  flow  in  rectangular  flumes  fre- 
quently indicate  values  of  the  coefficient  (a)  as  high  as  vf  for 
small  depths,  its  value  is  generally  equal  to  v\  when  the  mean 
hydraulic  radius  is  equal  to  one-fourth  the  width  of  the  channel. 
Its  value  is  less  than  the  normal  when  the  depth  exceeds  one -half 
the  mean  width  of  the  channel.  Applying  the  exponential  equa- 
tion 

=    log  Vt - log    ^o 

~  log  (66  (Sir  +  m)  Vrs\  -  log  (66  (Sir  +  m)  Vr  •  s)0 

to  data  relating  to  flow  in  a  semi-circular  channel  lined  with 
neat  cement  (Darcy-Bazin,  series  24)  we  find 

x  =  iJ  very  near- 

Applying  the  same  equation  to  data  relating  to  flow  in 
channels  lined  with  rough  boards,  semicircular  in  section,  we 
find  (Darcy-Bazin,  series  26) 

x  ^  it» 

thus  indicating  a  slight  decrease  in  the  value  of  a  with  increasing 
roughness  of  the  conduit's  wet  perimeter.  As  a  mean  between 
these  two  values  and  differing  but  slightly  from  either  we  may 
take 

*-« 

which  corresponds  to  a   =  V™ 

or  a  =\/66  (Vr  +  m)  Vrs 

and  the  frictional  resistance  is  proportional  to  V"*~  =  V  '  • 
This  value  of  the  power  of  the  velocity  we  observe  is  the  identical 
value  Froude  found  for  smooth  plain  surfaces  in  his  investiga- 
tions on  the  resistance  of  ships. 

Besides  the  two  series  mentioned,  given  by  Darcy-Bazin,  we 
find  that 

x  =  if 

holds  also  good  for  the  following : 


OPEN   CONDUITS  45 

Darcy-Bazin,  series  25,  semicircular  channel  lined  with  smooth 

concrete. 
McDougall,  Provo  Canal  Flume,  semicircular  channel  of  planed 

staves. 
Th.  Horton,  Conduit  of  North  Metropolitan  Sewage  System  of 

Massachusetts.     Brickwork  washed  with  cement.     Diameter 

9  feet.    Values  of  R  up  to  2.31  feet. 


F. 

Applying  the  experimental  equation  as  indicated  above  to 
data  relating  to  flow  in  channels  not  semicircular  in  section 
and  lined  with  cement  or  concrete,  planed  or  rough  boards, 
brickwork  and  good  ashlar  masonry  we  find 

x=  If 


a   = 


y 


66  (\'r  +  m)V 


rs 


and  the  frictional  resistance  is  proportional  to  F   .    This  we 
find  to  hold  good  for  the  following  : 

Darcy-Bazin,  series  2,  neat  cement,  section  rectangular. 
Darcy-Bazin,  series   6,  7,  8,  9,  10,  11,  18,  19,  21,  22,  and   23, 

sawed  boards,  section  rectangular,  triangular  or  trapezoidal. 
Darcy-Bazin  series  32,  33,  39,  channels  lined  with  good  ashlar 

masonry,  section  trapezoidal. 

Darcy-Bazin,  series  3,  rough  brick  work,  section  rectangular. 
Darcy-Bazin,  series  4,  channel  lined  with  pebbles  up  to  J  inch 

in  diameter,  section  rectangular. 
Fteley  and  Stearns,  Sudbury  conduit,  very  good  brickwork, 

sides  of  channel  nearly  vertical,  bottom  flat  arch. 
Fairlie  Bruce,  Aqueduct  of  Glasgow,  smooth  concrete,  sides  of 

channel  nearly  vertical,  bottom  flat  arch. 
Th.  Horton,  Conduit  of  North  Metropolitan  Sewage  System  of 

Massachusetts,  brickwork  washed  with  cement,  covered  with 

sewer  slime,  sides  of  conduit  vertical,  bottom  flat  arch. 


46  THE   FLOW   OF  WATER 

Lippincott,   San   Bernardino   Canal     Trapezoidal  channels   in 

earth,  lined  with  concrete. 
Kutter,  Gontenbachschale,  new  and  well  built  channel  of  dry 

rubble  masonry. 
Passim    and   Gioppi,  Aqueduct   of   the   Cervo,  Canal   Cavour. 

Floor  of  concrete,  sides  of  brick,  section  rectangular.    Values  of 

R  up  to  7.2  feet. 

G. 

Applying  the  exponential  equation  as  indicated  to  data  relating 
to  flow  in  channels  having  walls  possessing  a  greater  degree  of 
roughness  than  those  enumerated  above  we  find 

x  =  1.0 
a  =  1.0 

and  the  frictional  resistance  is  proportional  to  v2.    This,  amongst 
others,  holds  good  for  the  following : 

Darcy-Bazin,    series    1,  34,    35,   channels   lined   with   roughly 

hammered  stone  masonry. 
Darcy-Bazin,  series  5,  channel  lined  with  pebbles  1J  inch  to 

1J  inch  in  diameter. 

Kutter,  numerous  channels  lined  with  dry  rubble  masonry. 
Perrone,  Torlonia  drain  tunnel,  channel  in   rockwork,  partly 

lined  with  rubble  masonry. 

We  mention  here  also: 

Cunningham,  Aqueduct  of  the  Solani,  Ganges  Canal.     Floor  of 
brick,  laid  flat,  sides  of  masonry,  length  920  feet. 

In  this  case  the  fact  that  c  does  not  vary  with  the  velocity  of 
flow  is  due  to  the  shortness  of  the  conduit.  It  has  no  independent 
slope  and  the  movement  of  the  water  is  influenced  by  the  greater 
resistance  in  the  rough  channel  in  earth  downstream.  This  is 
plainly  indicated  by  the  low  value  of  the  coefficient  m. 

Of  open  conduits,  not  channels  in  earth,  there  are  few  possess- 
ing a  degree  of  roughness  still  greater  than  those  enumerated, 
exceptional  cases  of  old  and  damaged  rubble  masonry. 


CHANNELS   IN   EARTH  47 

H. 

The  variation  of  the  coefficient  c  with  the  velocity  of  flow  as 
deduced  from  experimental  data  relating  to  flow  in  open  conduits 
not  channels  in  earth  may  be  briefly  summarized  as  follows: 

1.  For  semicircular  channels  lined  with  cement,  concrete, 
good  brickwork,  planed  or  rough  boards,  the  value  of  the  coeffi- 
cient a  is  equal  to  Fn. 

2.  For  rectangular,  triangular  or  trapezoidal  channels  of  the 
same    description,    for   channels   lined   with  rough  brickwork, 
ashlar  and  very  good  rubble  masonry,  for  channels  lined  with 
pebbles  up  to  J  inch  diameter  the  value  of  the  coefficient  a  is 
equal  to  Fn. 

3.  For    channels    lined    with   roughly    hammered    stone    or 
common  rubble  masonry,  for  channels  lined  with  pebbles  up 
to  1^  inch  in  diameter,  for  channels  in  rockwork,  for  aqueducts 
of  any  description  discharging  into  channels  in  earth  and  having 
no  independent  slopes,  the  value  of  the  coefficient  a  is  equal  to  1.0. 

4.  For  channels  with  obstructions  producing  shocks,  such  as 
channels  with  cleats  nailed  crosswise  to  retard  the  flow,  for 
channels  lined  with  old  and  damaged  masonry  the  value  of  the 

coefficient  a  is  equal  to  — j _ 1  • 

Channels  in  Earth. 
I. 

When  we  scrutinize  the  data  relating  to  flow  in  rivers  and 
other  channels  in  earth  we  perceive  that  these  data  contain  many 
irregularities  and  contradictions  which  make  them  appear  doubt- 
ful and  untrustworthy.  Even  those  given  by  the  best  authori- 
ties are  not  entirely  free  from  anomalies.  These  irregularities 
and  contradictions  are  occasionally  the  result  of  inaccurate 
measurements;  more  often,  however,  they  must  be  attributed 
to  the  unstable  character  of  the  beds  of  these  channels.  This 
instability  of  the  bed  of  the  channels  makes  the  phenomenon 
of  flow  a  problem  of  great  complexity.  An  exact  valuation  of 
all  the  facts  entering  is  as  yet,  with  the  incomplete  data  at 
present  available,  out  of  the  question.  We  here  leave  the  path 


48  THE   FLOW   OF   WATER 

of  exactitude  and  enter  a  labyrinth,  satisfied  if  we  come  out  with 
the  gain  of  an  increment  of  knowledge  which  may  prove  useful. 

Natural  and  artificial  channels  in  rock  work  or  earth  may  be 
divided  according  to  the  stability  of  their  beds,  into  three  classes : 

1.  Channels  having  beds  in  a  regime  of  stability  at  velocities 
exceeding  the  ordinary.    Channels  in  rockwork,  cemented  gravel, 
channels  in  earth  protected  by  riprap  or  masonry  side  walls. 

2.  Channels  in  a  regime  of  stability  at  ordinary  velocities. 
Channels  in  gravel,  stiff  clay,  clayey  loam,  sandy  soils  with  over 
50  per  cent  clay. 

3.  Channels  in  a  regime  of  instability  at  ordinary  velocities. 
Channels  in  sand,  sand  with  fine  gravel,  sandy  loam  with  less 
than  50  per  cent  clay. 

The  beds  of  the  second  and  third  class  are  in  a  regime  of 
stability  until  the  velocity  becomes  sufficiently  great  to  erode 
the  bed. 

The  velocity  at  which  erosion  begins  varies  with  the  cohesion 
of  the  material.  In  channels  in  sand,  sandy  gravel,  sandy 
soils  with  small  percentages  of  clay,  erosion  begins  at  very  low 
velocities;  these  channels  are  consequently  very  unstable. 
Omitting  channels  in  firm  rock  or  cemented  gravel,  the  stability 
of  the  bed  depends  mainly  on  the  percentages  of  clay  in  the 
material.  According  to  W.  A.  Burr  pure  clay  resists  erosion  up 
to  a  velocity  of  7.35  feet  per  second.  The  following  table,  based 
chiefly  on  Burr's  experiments,  gives  the  mean  velocities  at  which 
erosion  begins: 


Nature  of  Material  Forming  the  Bed. 

Mean 
Velocity. 

Fine  sand     

0  72 

Coarse  sand,  sand  with  pebbles  up  to  pea  size  

1  10 

Sandy  soil  15  per  cent  clay      

1  20 

Fine  gravel  up  to  £  inch  in  diameter      

1  50 

Sandy  loam  45  per  cent  clay 

1  80 

Common  loam   65  per  cent  clay 

3  00 

Gravel  or  pebbles  from  ^  to  1  inch  in  diameter    . 

3  15 

Coarse  gravel      ...                   ....        

4  00 

Clayey  loam,  85  per  cent  clay  

4  80 

Clay  soil,  95  per  cent  clay,  loose  rock     

6  .20 

Stratified  rock  slaty  rock 

7  45 

Hard  rock 

12  00 

CHANNELS   IN   EARTH  49 

In  the  process  of  erosion  energy  is  consumed  which  varies 
with  the  specific  gravity  and  the  cohesion  of  the  material. 

The  erosive  power  of  a  current  is  proportional  to  the  square 
of  its  speed.  Its  transporting  power,  however,  varies  (according 
to  Le  Conte) : 

When  the  surface  is  constant  with  v2. 

When  the  velocity  is  constant  with  the  surface  of  the  object  or 
with  d\ 

When  both  vary  the  assistance  is  equal  to  v2  d2.  But  the 
weight  of  the  object  is  proportional  to'd3. 

Hence,  when  the  forces  are  in  equilibrium  or  the  weight  equal 
to  the  energy  d3  =  v2  d2. 

Dividing  by  the  surface  or  d2  we  have  d  =  v*. 

Consequently  when  the  forces  are  in  equilibrium  the  resistance 
is  proportional  to  v6.  In  other  words,  the  transporting  power 
of  a  current  is  proportional  to  the  sixth  power  of  the  speed. 
This  indicates  that  powers  of  r  ranging  between  2  and  6  enter 
the  problem  of  flow  when  erosion  begins. 

With  the  beginning  of  erosion  the  destruction  of  the  bed  will 
be  the  greater ;  the  less  the  cohesion  of  material  the  greater  the 
velocity.  Changes  and  alterations  in  course  and  section  generally 
continue  till  a  channel  is  formed  which,  owing  to  its  greater 
length,  its  deflections,  curves  and  bars  offers  such  resistances 
that  the  power  of  the  current  is  reduced  and  course  and  section 
again  become  stable  when  force  and  resistance  are  in  equi- 
librium. A  stream  will  pick  up  material  in  a  narrow,  deep 
section  of  its  course  where  the  force  of  the  current  is  great,  and 
deposit  it  in  a  wide  and  shallow  section  where  the  current  is 
feeble.  At  high  water,  the  greater  depth  of  the  water  in  the 
shallow  section  will  result  in  greater  velocities,  the  material 
previously  deposited  will  again  be  put  in  motion  and  carried  to 
a  place  where  the  current  is  feeble. 

The  work  done  during  these  processes  of  building  and  rebuild- 
ing cannot  be  accurately  measured,  and  on  this  account  slope 
formulae,  when  applied  to  flow  in  channels  where  erosion  is  going 
on,  are  always  more  or  less  deficient.  They  cannot  be  depended 
on  in  computing  discharges;  this  falls  into  the  province  of  the 


60  THE   FLOW   OF   WATER 

current  metre  and  the  rod  float.  They  are  useful,  however,  as  a 
guide  to  the  engineer  in  the  design  of  new  conduits,  alterations 
in  courses  or  sections,  etc.,  etc. 

The  banks  of  channels  having  unstable  beds  are  frequently 
protected  by  riprap  or  masonry  walls.  Frequently  the  bottoms 
of  such  channels  are  also  protected  by  artificial  bars  made  of 
boulders  or  masonry. 

Rittinger,  Borneman,  Epper,  Cunningham,  and  others,  have 
given  us  data  relating  to  flow  in  such  channels.  An  analysis 
of  these  data  gives  surprising  results.  Using  the  exponential 
equation 

x  =  log  vl  -  log  y0 
log  rt  -  log  r0 

we  find  the  following  values  of  x,  the  power  of  the  mean  hydraulic 
radius  to  which  the  velocity  is  proportional: 

Rittinger,  millrace  of  dry  rubble  side  walls,  bed  very  rough, 
depth  of  water  0.40  to  0.90,  x  =  3.0. 

Rittinger,  mill  race,  bed  sand  and  gravel,  side  walls  of  masonry, 
depth  0.28  to  0.90  ft.,  x  =  1.77. 

Rittinger,  Aqueduct  in  earth  lined  with  dry  rubble  side  walls, 
depth  0.61  to  1.27  feet,  x  =  1.19. 

Cunningham,  Solani  Embankment,  sides  of  masonry  built  in 
steps,  bed  of  clay  and  boulders,  with  frequent  artificial  bars 
to  prevent  erosion.  Main  site,  width,  150  to  170  ft.;  depth  of 
water,  1.7  to  4.1  ft.,  x  =  1.49;  depth  of  water,  5.6  to  9.34  ft., 
x  =  0.9;  Jaoli  site,  depth,  6.8  to  8.1  ft.,  x  =  0.93. 

Excluding  extremes,  the  powers  of  R,  to  which  the  velocity 
is  proportional  as  expressed  in  these  data,  may  be  given  by  the 
equation 

x   =  1.8     -  O.IR 

so  that  for  R  =  1.0  x  =  1.7 

R  =  2.0  x  =  1.6 
R  =  9.0  x  =  0.9. 

A  high  value  of  x  indicates  a  low  value  of  the  coefficient  c, 
but  a  rapid  increase  in  its  value  with  increasing  value  of  R]  a 


CHANNELS   IN   EARTH  51 

low  value  of  x  indicates  a  high  value  of  c  and  a  slow  increase  in 
its  value  with  increasing  values  of  R. 

The  influence  of  the  roughness  of  the  bed  is  necessarily  much 
greater  when  the  water  in  the  channel  is  shallow  than  when  it  is 
high;  the  diminishing  values  of  x  indicate  a  rapid  decrease  in  the 
relative  influence  of  the  character  of  the  bed.  But,  on  the 
other  hand,  while  the  powers  of  R  are  abnormally  high  for 
shallow  water  in  rough  channels,  the  powers  of  the  sine  of  the 
slope. to  which  the  velocity  is  proportional  are  abnormally  low. 
This  may  be  illustrated  by  data  deduced  from  experimental 
values  relating  to  flow  in  rough  channels  in  earth.  Amongst 
others  we  find: 

Wampfler,  Simme  Canal,  coarse  gravel  and  detritus, 

#1.104   £0.23> 

La  Nicca,  Rhine   in   the    Forest,    coarse  gravel  and   detritus, 

depth  0.42  to  0.9  feet.    R?'g  S  °'4. 
La  Nicca,  Plessur  River,  coarse  gravel  to  detritus,  depth  1.25 

to  4.58  feet.    #°'64  S0'4. 
Darcy-Bazin,  Grosbois  Canal,  Chazilly  Canal,  channels  in  earth, 

with  stones  and  vegetation,  depth  1.5  to  3.0  feet. 

#0.87        £0.43  to  #1.59        £0.4^ 

Reich,  River  Salzach,  gravel  and  detritus,  depth  3.53  to  7.39  ft. 

#0.8  £0.333^ 

Funk,  Weser  River,  depth  4.5  to  11  ft.    ft0'79  S0'5. 
Villevert,  River  Seine,  depth  5.66  to  18.39  ft.     R0'63  S°'443. 

In  general  therefore,  for  shallow  water  in  rough  channels  the 
power  of  the  sine  of  the  slope  to  which  the  velocity  is  proportional 
is  equal  to  0.4  and  equal  to  0.473  for  depths  exceeding  4  feet. 
The  variations  in  the  powers  of  both  r  and  S  with  the  depth  of 
the  water  in  the  channel  are  chiefly  due  to  the  fact,  that  the 
bottoms  of  such  channels  are  in  most  cases  much  rougher  than 
the  sides.  In  shallow  water,  the  resistance  due  to  the  bottom 
preponderates,  with  increasing  depth  the  influence  of  the  less 
rough  sides  more  and  more  reduces  the  mean  resistance  per 
unit  area  of  surface. 

The  powers  of  r  vary  not  only  with  the  degree  of  roughness 


52  THE   FLOW   OF   WATER 

in  general  and  with  the  depth  of  the  water,  but  also  with  the 
value  of  a,  the  coefficient  of  variation  of  c.% 

For  the  same  degree  of  roughness,  the  powers  of  r  have  their 
highest  value  for  the  highest  value  of  a. 

For  m  =  —  0.33  or  K  =  2.0  for  instance, 

and  a  =  1.0    R*  =  R0'795. 

But  for  a  =  V"  R-  =  R0'835 

for          a  =  A  R-  =  #9'745 

yrs 

for          a  =   A-  R*  =  R0'66. 

V* 

This  shows  the  great  influence  of  bends,  bars,  or  other  impedi- 
ments on  the  powers  of  R. 

Our  general  equation  expresses  the  variation  of  the  powers  of 
r  with  the  depth  with  a  fair  degree  of  accuracy.  Greater 
accuracy  is  obtained  if  the  formula  is  put  into  the  form 

c  =  66  f  \/r  +  ("2"  1  +  ^r  JJ  and  giving  m  a  negative  value,  as 

for  instance : 

for  K  =  1.20  m  =  -  0.10 

for  K  =  1.50  m  =  -  0.20 

for  K  =  2.0m    =  -  0.33. 

For  values  of  R  less  than  1.0  foot  the  formula 
66  (\/r  +  Vf) 

Vr+K 
gives  slightly  excessive  results. 

Amongst  the  mass  of  experimental  data  accumulated  during 
recent  years  those  given  by  Fortier  for  irrigation  channels  are, 
considered  from  the  practical  standpoint,  the  most  valuable. 
They  relate  to  flow  in  channels  possessing  all  possible  degrees 
of  roughness  and  a  minute  description  of  the  nature  of  the  bed 
is  always  given.  Gaugings  were,  however,  taken  only  for  a  single 
depth  and  a  single  slope  at  each  section  and  on  this  account 
no  deductions  can  be  made  in  regard  to  the  variation  of  the 
coefficient  c  with  the  velocity. 


CHANNELS   IN    EARTH  53 

Besides  these  Dubuat,  Darcy-Bazin,  Legler,  Cunningham, 
Rittinger  and  others  have  given  valuable  data  relating  to  flow  in 
canals  to  ditches;  Funk,  Villevert,  Revy,  Gordon  and  the  U.  S. 
Engineers,  interesting  data  relating  to  flow  in  rivers.  After  a 
careful  analysis  of  all  the  material  available  we  come  to  the 
following  conclusions  in  regard  to  the  variation  of  the  coefficient 
c  with  the  velocity: 

1.  For  channels  of  fairly  regular  cross-sections  and   courses 
having  tolerably  smooth  beds,  such  as  channels  in  firm  clay, 
clayey  loam,  sandy  soil  with  over  50  per  cent  clay,  fine  cemented 
gravel,  the  coefficient  c  increases  at  ordinary  velocities  with  the 
velocity  of  flow.     Under  ordinary  velocities  in  this  sense  we 
understand  velocities  which  do  not  cause  erosion. 

The  increase  in  the  value  of  c  with  increasing  velocities  is 
equal  to 

a  =  Vh 

for  the  smoothest  down  to 

a  =  V™ 

for  the  roughest  ohannels  of  this  class. 

Examples : 

S.  Fortier,  Bear  River  Canal  Branch. 
S.  Fortier,  Providence  Canal. 
S.  Fortier,  Solveron  and  Logan  City  Canals,  Utah. 
Darcy-Bazin,    rectangular  channel   lined    with    pebbles  up  to 

J  inch  diameter. 

Epper,  millrace,  channel  in  earth,  bottom  covered  with  fine  gravel. 
Dubuat,  Canal  du  Jard.     Channel  in  earth. 
Reich,  River  Salzach,  reach  very  regular. 

2.  At   velocities    exceeding    the  ordinary,  or  when  erosion 
begins,  the  coefficient  c  decreases  in  value  for  the  classes  of 
channels  enumerated   above.      The   decrease   is   usually  such 
that 

1 

a  =  —jrm 


54  THE   FLOW    OF  WATER 

Examples : 
Legler,  Linth  Canal.    The  coefficient  c  increases  until  v  is  equal 

to  4.72  ft.  per  second,  then  decreases. 
Gordon,  Irrawaddi  River.    The  coefficient  c  increases  until  v  is 

equal  to  2.62  ft.  per  second,  then  decreases. 

In  the  first  case  the  bed  is  firm  earth,  in  the  second  sand. 

3.  For  channels  of  fairly  regular  cross-section  and  course  in 
rock  work,  firm  gravel  up  to  2  inches  diameter,  for  channels  in 
firm  earth  or  sand,  or  sand  with  gravel,  with  stones  or  vegeta- 
tion, the  coefficient  c  does  not  appreciably  vary  with  the  velocity 
of  flow.     Consequently 

a  =  1.0. 
Examples: 

Perrone,  Torlonia  Drain  tunnel,  channel  in  rock  work. 
Darcy-Bazin,  series  5,  rectangular  channel  lined  with  pebbles  up 

to  1^-inches  diameter. 
Darcy-Bazin,  series  36,  37,  38,  41,  43,  47,  48,  50,  Grosbois  and 

Chazilly  Canals.    Channels  in  earth  of  regular  cross-section  but 

with  stones  or  weeds. 
La  Nicca,  Moesa  River,  coarse  gravel. 
La  Nicca,  Plessur  River,  coarse  gravel. 
Funk,  Weser  River. 
Passini  and  Gioppi,  Canal  Cavour,  below  the  Syphon  of  the  Sesia. 

4.  For  the  class  of  channels  enumerated  under  (3)  the  co- 
efficient c  decreases  in  value  whenever  the  velocity  becomes 
sufficient  to  cause  erosion.    The  decrease  usually  corresponds  to 

1 

a  =  — r  ' 
V* 

5.  For  channels  with  very  rough  beds,  channels  with  boulders, 
loose  cobblestones,  loose  coarse  gravel  or  detritus,  for  channels 
with  artificial  bars  to  prevent  scour,  the  coefficient  c  decreases 
rapidly  in  value  with  increasing  velocities.    The  decrease  is  equal 
to 

1 
a  =  — ?  • 

V* 


CHANNELS   IN   EARTH  55 

Example  : 

Cunningham,  Solani  Embankment,  bed  in  clay  and  boulders  with 
artificial  bars  to  prevent  erosion,  sides  of  masonry. 

Omitting  the  extremes,  we  may  briefly  sum  up  the  variation  of 
the  coefficient  c  with  the  velocity  as  follows: 

1.  For  channels  of  very  regular  cross-  sections  and  courses  in 
clay,  clayey  loam,  sandy  soils  with  large  percentages  of  clay, 
cemented  gravel  up  to  one  inch  in  diameter,  the  coefficient  of 
variation  of  c  is  equal  up  to  the  eroding  limit  to 


a  = 

2.  For  channels  in  rock  work  or  cemented  gravel  exceeding  one 
inch  in  diameter,  for  ordinary  channels  in  earth,  channels  with 
some  stones  or  vegetation,  the  coefficient  a  is  equal  up  to  the 
eroding  limit  to 

a  =  1.0. 

3.  For  channels  in  sand  at  any  velocity  and  for  all  others  at 
velocities  exceeding  the  eroding  limit,  the  coefficient  c  decreases 
in  value  with  increasing  velocities  and  the  coefficient  of  variation 
is  fairly  equal  to 

1 

a  =  —  j-  • 
vt* 


K. 

In  a  preceding  chapter  we  have  mentioned  the  experiments 
made  by  Hubbel  and  Fenkell,  Saph  and  Schoder  to  determine 
the  loss  of  head  due  to  the  resistance  in  curves.  From  data 

r> 

given  by  them  we  computed,  that,  omitting  values  of  -j  less 

(jL 

than  2.5,  the  friction  per  unit  length  of  curve,  in  terms  of  the 
friction  per  unit  length  of  tangent  is  equal  to 


56 


THE   FLOW   OF    WATER 


and  the  excess  of  friction  per  unit  length  of  curve  in  terms  of 
tangent  friction  is  equal  to 


and  the  length  of  tangent  equal  in  the  amount  of  frictional 
resistance  to  the  frictional  resistance  in  a  curve  of  90°  equal  to 


0.5  xR 


U.Qd*3 


^M  -  i.o. 


~R  7? 

This  vanishes  when  -y   =  4.9  3  d,  it   is   a   maximum   when  -r 
d  a 


=  4.9  3  d  and  the  total  excess  of   friction  is  greatest. 
loss  of  head  due  to  any  curve  is  consequently 


The 


)2r  2g 


TABLE  IV. 

FRICTION  IN  CURVES. 


2.5 
4 
5 
6 

10 
15 
20 
25 
50 
100 


2.5 
4 
5 
6 

10 
15 
20 
25 
50 
100 


Values  of  |4.9d*  f-1     ) —  1.0.     Diameters   1  to  72  Inches. 


I" 

2" 

4" 

6" 

12" 

18" 

24" 

30" 

36" 

48" 

60" 

72" 

0.375 

0.777 

1.422 

1.907 

2.971 

3.360 

3.657 

3.903 

4.113 

4.462 

4.750 

4.996 

0.271 

0.595 

1.174 

1.609 

2.564 

2.913 

3.080 

3.400 

3.588 

3.903 

4.160 

4.382 

0.225 

0.515 

1.065 

1.478 

2.386 

2.717 

2.971 

3.180 

3.359 

3.657 

3.903 

4.113 

0.188 

0.453 

0.980 

1.377 

2.247 

2.565 

2.808 

3.009 

3.180 

3.466 

3.701 

3.903 

0.091 

0.292 

0.761 

1.113 

1.887 

2.170 

2.388 

2.564 

2.717 

2.971 

3.186 

3.369 

0.020 

0.177 

0.604 

0.925 

1.630 

1.887 

2.084 

2.247 

2.386 

2.617 

2.808 

2.971 



0.101 

0.501 

0.802 

1.461 

1.703 

1.887 

2.039 

2.168 

2.386 

2.564 

2.717 

0.046 

0.426 

0.712 

1.337 

1.567 

1.742 

1.887 

2.010 

2.216 

2.386 

2.531 

0.216 

0.460 

0.994 

1.189 

1.338 

1.462 

1.567 

1.743 

1.887 

2.010 



0.037 

0.244 

0.700 

0.867 

0.994 

1.099 

1.189 

1.348 

1.461 

1.565 

Values  of  z.     Curve  of  90  degrees,     m  =  0.95. 

0.049 

0.091 

0.151 

0.184 

0.249 

0.258 

0.263 

0.266 

0.268 

0.270 

0.271 

0.272 

0.057 

0.112 

0.210 

0.248 

0.344 

0.358 

0.366 

0.371 

0.375 

0.378 

0.380 

0.382 

0.059 

0.121 

0.227 

0.285 

0.401 

0.416 

0.427 

0.432 

0.438 

0.443 

0.446 

0.448 

0.059 

0.128 

0.251 

0.319 

0.451 

0.472 

0.484 

0.492 

0.478 

0.504 

0.507 

0.510 

0.048 

0.137 

0.325 

0.429 

0.634 

0.666 

0.686 

0.699 

0.710 

0.720 

0.727 

0.734 

0.015 

0.147 

0.386 

0.535 

0.821 

0.869 

0.899 

0.918 

0.935 

0.951 

0.963 

0.971 

0.095 

0.427 

0.619 

0.982 

1.046 

1.085 

1.111 

1.133 

1.157 

1.173 

1.184 

0.054 

0.454 

0.687 

1.123 

1.203 

1.252 

1.296 

1.313 

1.343 

1.364 

1.379 

0.460 

0.888 

1.670 

1.825 

1.924 

2.001 

2.047 

2.113 

2.157 

2.190 

0.158 

0.931 

2.352 

2.836 

2.858 

2.995 

3.107 

3.268 

3.340 

3.410 

CHANNELS   IN  EARTH 


57 


TABLE  IV.    A. 

WEISBACH'S  COEFFICIENTS  FOR  RESISTANCES  DUE  TO  ENTRANCES,  ELBOWS, 
CURVES,  CHANGES  OF  SECTION,  ETC.,  ETC. 


Values  of  z. 

Description  of  Resistance. 

0.054 

Funnel-shaped  or  bell-mouthed  entrance  not  pro- 
truding into  the  reservoir. 

0.505 

Well   rounded   entrance  not   protruding  into   the 
reservoir. 

0.505 

Funnel-shaped  or  bell-mouthed  entrance  protruding 
into  the  reservoir. 

1.957 

Ordinary  pipe  protruding  into  the  reservoir. 

0.9457  sine2  1  + 
2.047  sine4^ 

Elbows  d  =  angle  of  deflection. 

o.i3i  +  uW^y 

Curves.     Section     circular,     d   =  diameter,     R  = 
radius  of  curve. 

0.124  +  3.104^^ 

Curves.     Section  rectangular,     d  =  Width  of  side 
parallel  to  R,  the  radius  of  the  bend. 

a-'J 

n         .    .                         Section  contracted 

Section  not  contracted 
a  =   1.225  +  1.45  w2  -  1.675m. 

(*-')• 

Enlargements  or  Contractions.      Al  —  Section  not 
contracted,  A2  =  Section  contracted. 

™& 

Bends  of  Rivers,     n  =  Number  of  degrees  in  arc  of 
bend. 

i-°B7i(^?-1) 

Obstructions  in  Rivers,     m  =  Percentage  not  ob- 
structed. 

vx  being  equal  to  V~* ,  V^~,  V™~,  etc.,  according  to  the  degree  of 
roughness  of  the  conduit.  The  coefficient  of  frictional  resistance 
is  given  by 


+  m)2r 


58  THE  FLOW   OF   WATER 

in  these  equations 

n  =  number  of  degrees  in  curve. 

TT  =  3.1416. 

d  =  diameter  of  conduit  in  feet. 

R  =  radius  of  curve  in  feet. 

x  =  -£$  for  diameters  greater  than  1  foot. 

x  =  0.45  for  diameters  less  than  1  foot. 

y  =  |  for  a  diameter  of  1  inch. 

y  =  A  f°r  any  °ther  diameter. 

From  the  foregoing  we  draw  the  conclusion,  that  the  value  of 
z  depends: 

r> 

1.  On  the  value  of  -7-  and  the  value  of  d. 

a 

n° 

2.  On  the  value  of  -       • 


3.   On  the  value  of  m. 

For  any  arc,  multiply  the  values  of  2,  found  in  the  table,  by 
the  number  of  degrees  and  divide  by  90. 

For  any  degree  of  roughness  multiply  the  values  of  z  by 
the  following: 

m  =  0.95,  multiply  by  1.0. 
m  =  0.83,  multiply  by  1.166. 
m  =  0.68,  multiply  by  1.436. 
m  =  0.53,  multiply  by  1.802. 
m  =  0.45,  multiply  by  2.060. 
m  =  0.30,  multiply  by  2.717. 

If  in  the  formula  for  the  loss  of  head  due  to  a  curve  we 
substitute 

2  grs  ,      ..          .     ,          0.01478 
—  —  for  its  equivalent  -r-=  - 
V2  (  Vr  +  m)2 

and  L  for  the  length  of  the  curve  the  formula  will  read,  after 
reduction, 


RIVETED   CONDUITS  59 

which  simply  expresses  the  theory  outlined  at  the  beginning  of 
this  chapter  that  the  excess  loss  of  head  due  to  a  curve  is 

/4.9  f  fCA  -  1.0 


times  the  loss  due  to  an  equal  length  of  straight  pipe;  S  being 
the  sine  of  the  slope  to  which  velocities  in  the  tangent  are  due. 

Riveted  Conduits. 

L. 

Riveted  conduits  form  a  class  apart  in  so  far  as  the  degree  of 
roughness  varies  with  the  diameter.  Up  to  date  the  coefficients 
for  such  conduits  have  been  fairly  well  determined  for  diameters 
up  to  8.5  feet  (Holyoke  Testing  Flumes);  for  larger  sections 
they  are  as  yet  problematical. 

Fairly  reliable  values  of  the  coefficients  for  riveted  conduits 
may  be  found  by  computing  the  losses  of  head  due  to  the 
resistance  of  rivet  heads,  or  to  enlargements  and  contractions 
of  the  section  as  follows : 

If  in  an  18-foot  steel-riveted  pipe  we  allow  an  internal  pressure 
of  140  pounds  per  square  inch,  in  the  steel  a  tension  of  20,000 
pounds  per  square  inch;  and  if  we  assume  the  efficiency  of  the 
riveted  joints  to  be  70  per  cent  of  the  metal,  we  have  for  the 
thickness  of  the  metal  in  inches 

140  X  diameter  in  inches, 
0.7  X  40,000 

which  gives  t  =  1.08  inches. 

It  is  usual  to  take  for  the  diameter  of  the  rivet  in  inches 

d  =  0.15  +  1.5  t, 
and  for  the  pitch  of  the  rivets  in  a  single  row 

sl  =  0.375  +  2  d, 
and  s2  =  0.75  +  3  d 

for  the  pitch  in  a  double  row. 


60  THE  FLOW  OF   WATER 

Hence  in  our  case 

d  =  1.75, 
s,  =  3.875, 
s2  =  6.0. 

The  usual  diameter  of  the  rivet  head  is  1.8  d  and  its  depth  0.6  d. 
This  gives  for  the  sectional  area  of  the  rivet  at  right  angles  to 
the  line  of  flow 

3.15  X  1.05  =  3.3075  square  inches  nearly. 

As  the  circumference  of  the  conduit  is  12  X  18  X  3.14  =  678.25 
inches  and  the  spacing  3.875  inches,  there  will  be  175  rivets  in 
the  single  circumferential  row.  The  open  space  between  the 
rivets  will  only  be  3.875  -  3.25  =  0.725  inches.  The  dis- 
turbance in  the  motion  in  this  narrow  space  will  be  such,  that 
it  will  be  safe  to  consider  the  row  of  rivet  heads  as  an  unbroken 
line  of  a  depth  0.6  d  =  1.05  inches.  Weisbach  gives  for  the  loss 
of  head  due  to  constrictions 


—-> 

2g 
in  which 

Al  =  section  not  constricted, 
A2  =  section  constricted, 


a  =  1.225  +      *     -  1.695       - 

\4J  Ai 

In  our  case       A,  =  182  X  0.7854  =  254.34, 

A2  =  (17.825)2  X  0.7854  =  249.5. 

Inserting  these  values  in  Weisbach's  formula  we  find 
h  =  00187489  ^-  - 

Assuming  the  metal  sheets  to  be  10  feet  each  way  there  will 
be  six  sheets  in  the  circumference,  and  as  the  pipe  is  double 
riveted  longitudinally  there  will  be  twelve  longitudinal  rows  of 
rivets,  and  allowing  1.6  d  for  the  outside  rim  on  each  side  there 


RIVETED   CONDUITS  61 

will  be  twenty  circumferential  rows,  the  pitch  being  six  inches. 
The  twelve  rivets  in  each  row  will  cause  a  constriction  of  12  x 
3.3075  =  39.69  square  inches  =  0.275  /2.  According  to  Weis- 
bach's  formula  this  constriction  causes  a  loss  of  head  equal  to 

h  =  00005936^-, 
and  the  twenty  rows  a  loss  equal  to 

h  =  0001  1907  ~ 

Adding  the  resistances  due  to  all  the  circumferential  rows  in  a 
section  of  9.5  feet  we  have 

Z,  =  00187489  +  00011907  =  00199396. 

Assuming  the  conduit  to  be  20,000  feet  long  the  total  resistance 
due  to  the  rivet  heads  will  be 

20000 
Z.=  -          =  2105  X  00199396  =  4.196985. 


To  this  must  be  added  the  resistance  due  to  the  enlargement 
or  contraction  caused  by  the  circumferential  lap  of  the  sheets. 
As  the  thickness  of  the  metal  is  1.08  inches  the  diameter  is 
enlarged  or  contracted  2.16  inches  at  each  lap.  The  loss  of 
head  due  to  enlargements  or  contractions  is,  according  to 
Weisbach, 


r   /     216"    \2  V  V2  V2 

hence  in  our  ease  [  (m^J     -  ij    ^  =  00041209  ^ 


The  total  resistance  due  to  all  the  enlargements  or  contractions 
is  consequently 

Z2  =  2105  X  00041209  =  0.86755. 

If  the  conduit  had  no  rivet  heads  or  enlargements  and  con- 
tractions to  increase  the  resistance,  the  value  of  the  coefficient  m 


62  THE  FLOW   OF   WATER 

would  be  the  same  as  for  a  cast-iron  pipe,  or  equal  to  0.83, 
and  the  frictional  resistance  per  unit  area  of  surface  would  be 

0.01478 
/=!  (1.456  +  0.83)2  ' 

and  the  total  resistance  of  the  wet  perimeter 

Z3  =  002829      ,  ..       =  12.473. 
4.5 

Adding,  we  have  for  the  sum  of  all  the  resistances 
Zl  +  z2  +  z3  =  17.5375. 

This  gives  for  the  total  frictional  resistance  per  unit  area  of 
surface 

17.5325  x  ^ 

or  /  =  00394594; 

hence  the  coefficient  c  is  equal  to  y          '        =  127.7,  and  m  is 

(  M  )§ "iVy^-rOr/  / 


127  7 
equal  to   -^ 1.456  =  0.48. 

OD 


Practical  Applications  of  the  Formulae 

M. 
1.   From  the  formula 

v  =  (66  (Vr  +  m) 
we  have 


and 

We  have  also 


—  1 

V              -i' 

v     _ 

^66(^r+ra)  Vr  ] 
Vr  +  m)Vr  =  Rl  4-  m  Vr. 

- 
66  Vs 


PRACTICAL  APPLICATIONS   OF  THE   FORMULAE  63 

Putting  'Vr  =  x  and  transposing  we  have 

X3  +  mX2  +  0 5-^—-=-  =  0, 

"  *-  66  Vs 


from  which  the  value  of  x  =  *vV  is  found  by  Horner's  method. 
We  have  also 


m  = 


66 


If  the  coefficient  of  variation  of  c  is  equal  to  7^,  V^*  —  j-  etc., 
these  values  are  substituted  in  the  given  equations. 
Values  of    a  =V%  7*,   7*    y^   y&    yrV 


> 

are  found  in  Table  V. 

Example:  Let  it  be  required  to  find  the  slope  for  a  rectangular 
aqueduct  of  common  brickwork  or  concrete  100  feet  wide,  12.5 
feet  deep,  the  velocity  to  be  4  feet  per  second.  The  cross-  sec- 
tion is  1,250  /2,  the  wet  perimeter  125  /,  hence  R  =  10.0.  In 
the  table  of  roots  of  mean  radii  we  find  \/10  =  3.163  VlO  = 
1.78.  The  value  of  m  for  common  brickwork  or  concrete  is 
0.57.  The  value  of  a  =  7T*  for  v  =  4.0  is,  according  to  Table  V, 
equal  to  1.08.  Inserting  these  values  into  our  formula  we  have 
for  the  slope 

r  _  4  _  7 

Ll.08  X  66  X  (1.78  +  0.57)   X  3.163J 


V530 
=  0.0000569. 

Example:  Let  it  be  required  to  find  the  diameter  of  a  semi- 
circular channel  lined  with  common  ashlar  or  very  good  rubble 
masonry,  the  slope  being  1  in  1,000,  and  the  permissible  velocity 
10  feet  per  second. 

In  this  case  m  =  0.30 

V7  =  0.0316 
a  =  ^To  =  1.137. 


64 


THE   FLOW   OF   WATER 


Solving  by  Homer's  method  and  inserting  values  we  have 


X3  + 


-  -        1.187  XMX0.0316 

0.3  X2  +  0.0  -  4.217  =  0.0  \c  =  1.521 
1.0     +  1.3  +  1.300 


-0"0 


1.3 
1.0 

+  1.3  -  2.917 
+  2.3 

2.3 
1.0 

+  3.6 

3.3 
0.5 

+  3.6 
+  1.9 

-  2.917 
+  2.750 

3.8 

0.5 

+  5.5 
+  2.15 

-  0.167 

4.3 
0.5 

+  7.65 

4.8 
0.02 

+  7.65 
+  0.096 

-  0.167 
+  0.1544 

4.82   +  7.746  -  0.0121 

0.001  +  0.005  +  0.0077 

4.821  +  7.751  -  0.0044. 

This  gives  x  =  1.521;   hence  the  mean  hydraulic  radius  r 
(1.521)4  =  5.352,  and  the  diameter  =  21.408  ft. 

2.   From  the  formula 

2gh  1ft 

0.01478     L 
tlr  +  m) 2  ~R 


v  = 


we  have 


2gh 


0.01478     L 


„_     0.01478     L  r 

—        4/— \~2    D  ~O 

L  V™ 

+  m)2  R  =  0.01478  -=7  ^— 
/i    z  g 


m 


)Vr  =  R*  +  mVr  -  V  0.01478 


— 


PRACTICAL   APPLICATIONS   OF  THE   FORMULAE  65 

and  putting  x  =  t/r  we  have 


ho-yo.r1"*"  L 


X3  +  m  X2  +  0.0  -  V  0.01478    ^  —  =  0.0, 

a.     2  g 

which  may  be  solved  by  Homer's  method. 
To  facilitate  calculations  it  is  well  to  remember 

that  y&=7*x  7*. 


and  7V-/4Y 

V7V 

Values  of  7  *  and  7^  are  found  in  Table  V. 

Resistances  due  to  entrance  and  the  velocity  itself  are  included 

in  the  term  —^= and  need  not  be  further  considered 

(Vr  4-  m)2  R 

unless  the  length  of  the  conduit  is  less  than  1,000  diameters. 
For  pipes  between  300  and  1,000  diameters  in  length  (as  also 
for  riveted  pipes  exceeding  3  feet  in  diameter),  the  coefficient  of 
variation  is  equal  to  a  =  7^,  and  A  and  V  are  substituted  in 
the  given  equations  for  A  and  V.  If  the  pipe  is  between  100 
and  300  diameters  in  length  (or  an  old  pipe  not  very  clean)  the 
coefficient  a  is  equal  to  1.0,  and  %  and  2.0  are  substituted  in 
the  given  equations  for  A  and  V6-  In  case  the  conduit  is  less 

than  100  diameters  in  length  the  coefficient  a  is  equal  to  — j-  and 
A  and  V  are  substituted  for  T9<r  and  V.  Values  of  °-01478 


are  found  in  Table  VI. 

Example:  Let  it  be  required  to  find  the  velocity  of  flow  in  a 
new  steel  riveted  conduit  6  feet  in  diameter,  10,000  feet  long, 
the  head  to  be  5  feet  and  the  conduit  to  have  20  curves  of  10° 
each  and  a  radius  of  30  feet.  In  this  casern  =  0.53,  R  =  1.5, 

Vl.5  =  1.107.    For  the  curves  we  have  the  relation  —  =  — - 

a        o 

r> 

=  5.0.    In  Table  IV  we  find  the  coefficient  z2  for  -  =  5.0  and  a 
curve  of  90°  to  be  equal  to  0.466.    As  there  are  20  curves  of  10 

degrees  we  have  Z2  =  2Q  X  ™X  °'448  -  0.995.    For  m  =  0.53 

yu 


66  THE  FLOW  OF   WATER 

this  is  to  be  multiplied  by  1.802,  which  gives  for  the  total  resist- 
ance due  to  curves  Z2  =  1.782. 

Inserting  values  into  our  formula  we  have 

64.4  X  5.0 


v  = 


0.01478  X  10000 


J1.107  +  0.53)2X  1.5 
=  (8.312)* 

Remembering  that  V*7  =  7*  X  V&  we  first  draw  the  square 
root  out  of  the  quotient  and  multiply  this  by  the  seventeenth 
root  of  the  square  root. 

The  quotient  is  8.312,  V8312  =  2.884.  In  the  table  of  roots 
we  find  ^3  =  1.065,  X/2.75  ••=  1.059.  Interpolating  we  have 
for  *!J 2^884  =  1.062.  Consequently!;  =  2.884  X  1.062  =  3.0628 
feet  per  second. 

3.   From  the  formula 


v  =  (66  (r  +  m) 

we  have  for  the  discharge  of  a  circular  conduit  in  cubic  feet  per 
second 

Q  =  (66  (j/r  +  m) V77)  *  d2  0.7854. 

From  this  we  have  for  the  head  in  feet 

«_y_ -i Ti. 

0.7854/   66  ($r  +  m)J  R 
and  for  the  diameter  in  feet 


).7854/     [66(-Vr  +  m)]2  H 

If  a  =  7T*  the  index  if  is  substituted  for  |,  if  for  f ,  V  for  V6 
and  A  for  49T-  From  this  equation  the  value  of  d  can  only  be 
found  by  trial,  assuming  a  value  of  $lr  in  the  term  ^Ir  +  m. 
For  a  first  trial  a  value  of  $r  =  1.0  will  give  good  results. 

From  the  formula 

2gH 


0.01478      L 

f     4/  \      o         -W-*L 

.vr  +  m)2  R 


PRACTICAL  APPLICATIONS   OF  THE   FORMULAE  67 


we  have         Q 


2gH 


0.01478      L 

( Vr  +  m) 2    R 


d2  0.7854 


0.01478     L 

A.  i —  I 


H_(  _  __\          r  + 
\  d2  0.7854/ 


2g 


0-01478      ,       7p,7  p" 
'  * 


From  this  equation  also  d  can  only  be  found  by  a  second  or 
third  trial,  assuming  a  value  of  Vr  and  R. 

For  a  =  FT*  the  index  T97  is  substituted  for  tV,  V  for  V  and 
A  for  49T- 

For  a  =  1.0  the  index  £  is  substituted  for  y9s,  2  for  V  and 
A  =  t  for  A. 

Example:  What  will  be  the  loss  of  head  corresponding  to  a 
discharge  of  5  cubic  feet  per  second,  the  conduit  being  a  2-foot 
riveted  pipe  20,000  feet  long  and  having  30  curves  of  15°  each 
and  a  radius  of  20  feet. 

In  this  case  m  =  0.68;  R  =  0.5;   Vr  =  0.84,  consequently 

0.01478  0.01478 


(Vr+TO)2       (0.84  +  0.68) 2 


In  Table  IV  we  find  for  the  resistance  of  a  90°  curve  for  the 

7?          OC\ 

relation   —  =  —  -  =  10    Z2  =  0.686,  consequently  for  30  curves 

d        — 

of  15°  each  Z2  -  30  X  15  X  °'686  -  3.430.    For  m  =  0.68  this 

yo 

is  to  be  multiplied  by  1.436  which  gives  z2  =  4.925. 
Inserting  these  values  we  have 


5  [0.00638  X  --     +  4.925 

L4  X  0.7854J  64.4 


or  H  =  9.24  feet. 


68  THE   FLOW   OF   WATER 

5.  The  Kinetic  energy  or  living  force  acquired  by  a  body 
falling  free  or  descending  in  a  plane  infinitely  smooth  is  equal  to 

E  =  \rntf  =  Q.W.H., 

Weight 

m  =  the  mass  of  a  body  =  -=  —  ^—  , 

Gravity 

Q  =  the  discharge  in  cubic  feet  per  second, 
W  =  the  weight  of  one  cubic  foot, 
H  =  the  total  fall  in  feet. 

Expressed  in  horsepowers  the  energy  is  equal  to 


or,  in  kilowatts,  to 

K.W. 


If  a  body  of  water  is  not  falling  free,  the  total  head  is  reduced 
by  an  amount  which  depends  on  the  velocity,  the  length  of  the 
conduit,  its  diameter  and  its  degree  of  roughness. 
The  loss  of  head  is  equal  to 


V2     0.01478 


2g        2g      2  g  (t/r  +m)2   R 

as  the  case  may  be.  For  conduits  of  equal  length  the  loss  is 
evidently  the  least  for  the  greatest  diameter  and  for  the  lesser 
speed  of  flow. 

For  a  given  diameter  the  efficiency  of  a  conduit  as  a  trans- 
mitter of  energy  is  greatest  when  the  speed  of  flow  is  such,  that 
one-third  of  the  total  available  head  is  consumed  in  overcoming 
frictional  resistances  (see  "  Adams  and  Gummel,"  Eng.  News, 
May  4,  1893)  . 

Example:  A  new  four  foot  steel  riveted  conduit  2,000  feet 
long,  under  a  head  of  300  feet  is  to  deliver  water  to  the  gener- 
ator at  such  a  velocity  that  its  efficiency  will  be  a  maximum. 
What  will  be  the  discharge  and  the  horsepower  transmitted? 


PRACTICAL  APPLICATION   OF   THE  FORMULAE.  69 

Allowing  one-third  of  the  total  head  to  be  spent  in  overcoming 

100 
frictional  resistances  we  have  v  =  =  0.05.     For  this  value 

zooo 

of  v  the  velocity  will  be 

v  =  (66  (1  +  0.53)*  Vl  X  0.05)  &  =  2.369  feet  per  second; 

the  discharge,   Q  =  2.369  X  162  X  |j  =30.4  cubic  feet  per  second; 

„  r>      30.4  X  62.4  X  200      _ftQ  n 

the  horsepower,  H  P  = — =  708.0. 

o5u 


TABLE  V. 

Table  V  contains  roots  of  velocities  or  values  of  (a),  the 
coefficient  of  variation  of  c. 

To  find  the  value  of  c  corresponding  to  any  velocity  multiply 
the  value  of  66  (t/r  +  m)  by  the  value  of  (a)  =  7*,  7T\ 

— r-.as  the  case  may  be. 

1 


To  find  the  velocity  multiply  the  value  of  66  (^r  4-  m)  Vr.s 
by  the  value  of 

(66  ( Vr  +  m)  VTJS)  *     which  in  Table  V  is  given  as  7*, 
(66  ( Vr  +  rn)  Vr.s) "   which  in  Table  V  is  given  as 
(66  ( Vr  +  m)  Vr.s) T7   which  in  Table  V  is  given  as 

T-= .     which  in  Table  V  is  given  as  — 

(66  ( Vr  +  m)  Vr.s)  A 

as  the  case  may  be. 
Also  ,  to  find  the  velocity,  multiply  the  value  of 

which  in  Table  V  is  given  as 


f- 
'  R 


70 


THE   FLOW   OF  WATER 


TABLE  V. 

ROOTS  OF  VELOCITIES  OR  VALUES  OF  (a),  THE  COEFFICIENT  OF 
VARIATION  OF  c. 


V 

V* 

V\ 

V* 

yfs 

yk 

yh 

jrA 

1 

FA 

1 

yr& 

0.25 

0.841 

0.857 

0.882 

0.891 

0.917 

0.925 

0.925 

1.081 

1.075 

0.50 

0.918 

0.925 

0.939 

0.944 

0  .958 

0.959 

0.961 

1  .040 

1.037 

0.75 

0.964 

0.964 

0.974 

0.976 

0.982 

0.981 

0.982 

1.018 

1.015 

1.0 

1.0 

1.0 

1.0 

1.0 

1.0 

1  .0 

1.0 

1.0 

1.0 

1.25 

1  .026 

1.025 

1  .0205 

.019 

1.013 

.013 

1.013 

0.987 

0.989 

1.50 

1  .052 

1.046 

1.038 

.034 

1.025 

.024 

.022 

0.978 

0.979 

1  .75 

1  .072 

1.064 

1  .052 

.048 

1.035 

.033 

.031 

0.97 

0.971 

2.0 

1  .090 

1.080 

1.065 

.0594 

1  .044 

.042 

.039 

0.962 

0.964 

2.25 

1  .107 

1  .095 

1.077 

.070 

1  .052 

.049 

1  .046 

0.956 

0.959 

2.50 

1.121 

1.107 

1  .087 

1  .080 

1.059 

.056 

1.052 

0.95 

0.953 

2.75 

1  .135 

1.118 

1.096 

1.086 

.065 

.061 

1.058 

0.945 

0.948 

3.0 

1  .147 

1.130 

1.105 

1.096 

.071 

.067 

1.063 

0.940 

0.943 

3.25 

.159 

1.140 

1.113 

1.103 

.076 

.072 

.068 

0.936 

0.940 

3.50 

.170 

1.150 

1  .121 

1  .110 

.081 

.077 

.072 

0.932 

0.936 

3.75 

.18 

1.158 

1.128 

1.116 

.086 

1.081 

.076 

0.929 

0.933 

4.0 

.189 

1  .166 

1  .134 

1.123 

.090 

1.085 

.080 

0.926 

0.930 

4.25 

.198 

1  .174 

1.141 

1  .128 

.094 

1.088 

.083 

0.923 

0.927 

4.50 

.207 

1  .182 

1  .146 

1  .133 

1  .098 

1  .093 

.087 

0.919 

0.924 

4.75 

1  .215 

1  .189 

1  .152 

1  .139 

1.102 

1  .096 

.090 

0.917 

0.921 

5.0 

1  .223 

1  .196 

1.158 

1.144 

1.106 

1  .10 

.093 

0.914 

0.919 

5.25 

1  .231 

1.203 

1.163 

1  .149 

1.109 

1  .103 

.098 

0.912 

0.917 

5.50 

1  .237 

1.209 

1.168 

1.153 

1  .112 

1.106 

.099 

0.910 

0.914 

5.75 

.244 

1.215 

1.173 

1.159 

1  .115 

1.108 

.102 

0.907 

0.912 

6.0 

.251 

1  .220 

1.177 

1.161 

1  .118 

1.110 

.104 

0.905 

0.910 

6.25 

.258 

1  .226 

1.181 

1  .165 

1  .121 

1  .114 

.107 

0.903 

0.908 

6.50 

.262 

.231 

1.186 

1  .169 

1.123 

1  .116 

.109 

0.901 

0.906 

6.75 

.269 

.237 

I  .190 

1  .173 

1.126 

.119 

.112 

0.899 

0.904 

7.0 

.275 

.241 

1  .194 

1  .176 

.129 

.121 

.114 

0.897 

0.902 

7.25 

.281 

.246 

1.197 

1.18 

.131 

.124 

.116 

0.895 

0.901 

7.50 

.286 

1.251 

1.201 

1.183 

.134 

.126 

1.118 

0.894 

0.899 

7.75 

.292 

1  .256 

1.205 

1  .186 

.137 

.128 

1.12 

0.893 

0.898 

8.0 

1  .297 

1  .260 

1  .208 

1.189 

.139 

.130 

1  .122 

0.891 

0.896 

8.25 

1.302 

1  .264 

1.212 

1  .192 

.141 

.132 

1  .124 

0.889 

0.895 

8.50 

1.307 

1  .268 

1.215 

1  .195 

.143 

.134 

1  .126 

0.888 

0.893 

8.75 

1  .311 

1  .272 

1.218 

1.198 

.145 

.136 

1.128 

0.886 

0.892 

9.0 

1  .316 

1.277 

1  .221 

1.201 

.147 

.138 

1  .130 

0.885 

0.891 

9.25 

1  .320 

1.280 

1  .224 

1  .204 

.149 

.140 

1.131 

0.884 

0.889 

9.50 

1.325 

1  .284 

1  .227 

.207 

.151 

.142 

1.133 

0.882 

0.888 

9.75 

1.329 

1.288 

1  .230 

.209 

.153 

.143 

1  .135 

0.881 

0.887 

10.0 

1.333 

1  .292 

1  .233 

.212 

1.155 

.145 

1  .137 

0.880 

0.886 

10.5 

1  .341 

1  .298 

1.238 

.216 

1.158 

.149 

1  .139 

0.877 

0.884 

11  .0 

1  .348 

1  .305 

1.244 

.221 

1  .161 

1  .152 

1  .142 

0.875 

0.881 

11  .5 

1  .357 

1  .310 

1  .249 

.226 

1  .165 

1  .155 

1  .145 

0.873 

0.879 

12.0 

1  .364 

1.318 

1  .254 

.230 

1  .169 

1  .158 

.148 

0.871 

0.877 

14.0 

1.39 

1.340 

1  .271 

.246 

1  .179 

1.168 

.158 

0.863 

0.870 

16.0 

1  .414 

1.365 

1  .286 

.260 

1  .189 

1  .178 

.169 

0.855 

0.864 

18.0 

.435 

1  .38 

1  .301 

.272 

1  .199 

1.185 

.175 

0.851 

0.859 

20.0 

1.450 

1.395 

1  .311 

.288 

1.204 

1  .193 

.182 

0.846 

0.854 

PRACTICAL  APPLICATIONS   OF    THE   FORMULA 


71 


TABLE  VI. 

VALUES  OF  66  (^r  +  m)  AND  CORRESPONDING  VALUES  OF  /,  THE 
COEFFICIENT  OF  FRICTION.     CONDUITS  UNDER  PRESSURE. 


sj 

§3 

m  =  0.95 

m  =  0.83 

m  =  0.68 

m  =  0.53 

TO  =  0.45 

TO  =  0.30 

S-S 

S£ 

c 

/ 

c 

/ 

c 

/ 

c 

/ 

c 

/ 

c 

/ 

i 

87.3 

00839 

79.9 

01007 

70.0 

01313 

60.1 

01780 

54.8 

02142 

44  .9 

03190 

2 

92.4 

00750 

84.5 

00901 

74.6 

01156 

64.7 

01534 

59.4 

01823 

49.5 

02625 

3 

95.7 

00699 

87.8 

00834 

77.9 

01060 

68.0 

01391 

62.7 

01636 

52.8 

02307 

4 

97.1 

00679 

90.2 

00790 

80.3 

00984 

70.4 

01298 

65.1 

01528 

55.2 

02111 

6 

102.0 

00615 

94.1 

00726 

84.2 

00907 

74.3 

01166 

69.0 

01351 

59.1 

01841 

8 

104.9 

00581 

97.0 

00684 

87.1 

00848 

77.2 

01080 

71.9 

01244 

62.0 

01673 

10 

106.3 

00566 

99.4 

00651 

89.5 

00803 

79.6 

01015 

74.3 

01165 

64  .4 

01551 

12 

109.4 

00535 

101  .5 

00624 

91  .6 

00767 

81.7 

00964 

76.4 

01102 

66.5 

01454 

14 

111  .2 

00517 

103.3 

00603 

93.4 

00737 

83.5 

00922 

78.2 

01052 

68.3 

01379 

16 

112.9 

00502 

105.0 

00583 

95.1 

00711 

85.2 

00886 

79.9 

01007 

70.0 

01313 

18 

114.4 

00489 

106.5 

00567 

96.6 

00689 

86.7 

00856 

81.4 

00971 

71.5 

01258 

20 

115.9 

00498 

107.8 

00554 

97.9 

00671 

88.0 

00829 

82.7 

0094 

72.8 

01214 

22 

117.0 

00467 

109.1 

00540 

99.2 

00654 

89.3 

00807 

84.0 

00912 

74.1 

01171 

24 

118.2 

00458 

110.3 

00528 

100.4 

00638 

90.5 

00785 

85.2 

00886 

75.3 

01134 

26 

119.4 

00449 

111  .5 

00517 

101  .6 

00623 

91.7 

00765 

86.4 

00862 

76.5 

0110 

28 

120.4 

00441 

112.5 

00507 

102.4 

00611 

92.8 

00747 

87.5 

00840 

77.6 

01068 

30 

121  .4 

00434 

113.5 

00499 

103.6 

00599 

93.7 

00733 

88.4 

00823 

78.5 

01044 

32 

122.2 

00428 

114.3 

00492 

104.4 

00590 

94.5 

00720 

89.2 

00808 

79.3 

01023 

34 

123.1 

00422 

115.2 

00485 

105.3 

00580 

95.4 

00707 

90.1 

00792 

80.2 

00100 

36 

124.0 

00416 

116.1 

00477 

106.2 

00570 

96.3 

00693 

91.0 

00777 

81  .1 

00978 

38 

124.9 

00410 

117.0 

00470 

107.1 

00560 

97.2 

0068 

91.9 

00762 

82.0 

00957 

40 

125.7 

00405 

117.8 

00464 

107.9 

00552 

98.0 

0067 

92.7 

00748 

82.8 

00938 

42 

126.4 

0040 

118.5 

00458 

108.6 

00545 

98.7 

0066 

93  .4  00737 

83.5 

00923 

44 

127.0 

00396 

119.1 

00453 

109.2 

00539 

99.3 

00651 

94.2 

00728 

84.1 

00909 

46 

127.9 

00391 

120.0 

00446 

110.1 

00531 

100.2 

00641 

94.9 

00714 

85.0 

00890 

48 

128.4 

00386 

120.8 

00441 

110.9 

00523 

101.0 

00631 

95.7 

00702 

85.8 

00874 

50 

129.4 

00382 

121  .5 

00436 

111  .6 

00516 

101  .7 

00622 

96.4 

00692 

86.5 

0086 

52 

130.0 

00379 

122.1 

00431 

112.2 

00511 

102.3 

00614 

97.0 

00684 

87.1 

00848 

54 

130.7 

00375 

122.8 

00426 

112.9 

00504 

103.0 

00606 

97.7 

00674 

87.8 

00834 

56 

131  .2 

00372 

123.3 

00423 

113.4 

00500 

103.5 

00600 

98.2 

00667 

88.3 

00826 

58 

131  .9 

00368 

124.0 

00418 

114.1 

00494 

104.2 

00592 

98.9 

00657 

89.0 

00812 

60 

132.5 

00364 

124.6 

00414 

114.7 

00489 

104.8 

00586 

99.5 

00650 

89.6 

00801 

62 

133.1 

00361 

125.2 

00410 

115.3 

00484 

105.4 

00579 

100.1 

00642 

90.2 

00791 

64 

133.6 

00358 

125.7 

00407 

115.8 

00480 

105.9 

00573 

100.6 

00636 

90.7 

00782 

66 

134.2 

00355 

126.3 

00403 

116.4 

00475 

106.5 

00567 

101  .2 

00628 

91  .3 

00772 

68 

134.7 

00352 

126.8 

00400 

116.9 

00471 

107.0 

00562 

101  .7 

00622 

91  .8 

00763 

70 

135.2 

00350 

127.3 

00397 

117.4 

00467 

107.5 

00557 

102.2 

00616 

92.3 

00755 

72 

135.8 

00347 

127.9 

00393 

118.0 

00461 

108.1 

00551 

102.8 

00609 

92.9 

00745 

'  78 

137.2 

00340 

129.3 

00385 

119.4 

00451 

109.5 

00536 

104.2 

00592 

94.3 

00723 

84 

138.5 

00333 

130.6 

00374 

120.7 

00441 

110.8 

00524 

105.5 

00574 

95.6 

00704 

90 

139.9 

00327 

132.0 

00369 

122.1 

00431 

112.2 

00511 

106.9 

00563 

97.0 

00684 

96 

141  .2 

00321 

133.3 

00362 

123.4 

00423 

113.5 

00499 

108.2 

00549 

98.3 

00666 

102 

142.2 

00316 

134.3 

00356 

124.4 

00416 

114.5 

00490 

109.2 

00539 

99.3 

00652 

108 

143.2 

00312 

135.3 

00351 

125.4 

00409 

115.5 

00482 

110.2 

00530 

100  .3 

00639 

114 

144.5 

00306 

136.6 

00344 

126.7 

00401 

116.8 

00472 

111  .5 

00518 

101  .6 

00624 

120 

145.7 

00301 

137.8 

00339 

127.9 

00393 

118.0 

00462 

112.7 

00506 

102.8 

00609 

126 

146.7 

00297 

138.8 

00334 

128.9 

00387 

119.0 

00454 

113.7 

00497 

103.8 

00597 

132 

147.7 

00293 

139.8 

00329 

129.9 

00381 

120.0 

00446 

114.7 

00489 

104.8 

00590 

138 

148.6 

00290 

140.7 

00325 

130.8 

00376 

120.9 

00440 

115.6 

00481 

105.7 

00576 

144 

149.5 

00286 

141  .6 

00321 

131  .7 

00371 

121  .8 

00434 

116.5 

00474 

106.6 

00566 

156 

151.4 

00279 

143.5 

00313 

133  .6 

00360 

123.7 

0042 

118.4 

00459 

108.5 

00557 

168 

153  .0 

00273 

145.1 

00305 

135.2 

00352 

125.3 

00410 

120.0 

00447 

110.1 

00546 

180 

154.6 

00268 

146.7 

00299 

136  .8 

00343 

126.9 

00400 

121  .6 

00435 

111  .7 

00538 

72 


THE   FLOW   OF  WATER 


TABLE  VI.    A. 

WELDED   PIPES. 

TUBES    OF     BRASS,    GALVANIZED   IRON,    SHEET  IRON,   STEEL,    ETC. 


1 

£    • 

Actual 

Values  of  66  (  ^r  +  ra)  Vr 

/— 

Loss  of  Head  in  Feet  per 
Unit  Length  of  Conduit 

ss 

So 

Diam- 

Area = 

=  c  Vr 

at  Unit  Velocity. 

1-1  S 

Is 

eter 
in 

d20.7854 

feet. 

m  = 

m  = 

m  = 

m  = 

m  = 

m  = 

m  = 

m  = 

& 

0.95 

0.83 

0.68 

0.45 

0.95 

0.83 

0.68 

0.45 

t 

0  .0225 

0  .00038 

6.058 

5.467 

4.722 

3.584 

1  .714 

2.154 

2.888 

5.025 

0  .0303 

0  .00072 

7.111 

6.423 

5.56 

4.239 

1  .274 

1  .561 

2.083 

3  .585 

.| 

0.0411 

0  .00133 

8.483 

7.665 

6.677 

5  .139 

0  .8949 

1  .096  * 

1  .444 

2.439 

I 

0  .0516 

0  .00209 

9.646 

8.745 

7.623 

5.398 

0.692 

0.842 

1  .108 

1  .850 

I 

0  .0686 

0  .00370 

11.34 

10.31 

9.006 

7.018 

0  .5008 

0  .6063 

0.794 

1  .308 

i 

0  .0873 

0  .00599 

13.02 

11.85 

10.38 

8.142 

0  .3801 

0  .4587 

0.597 

0  .9496 

U 

0.1150 

0  .01039 

15.22 

13.90 

12.22 

9.646 

0  .2781 

0  .3331 

0.431 

0.692 

li 

0  .1341 

0.01412 

16.65 

15.20 

13.39 

10.61 

0  .2268 

0  .2790 

0.359 

0.572 

2 

0  .1722 

0  .02339 

19.24 

17.59 

15.54 

12.40 

0.1740 

0  .2081 

0.267 

0  .4155 

2^ 

0  .2056 

0  .03320 

21  .34 

18.23 

17.30 

13.85 

0.1414 

0  .1936 

0.215 

0  .3354 

3 

0  .2556 

0  .05130 

24.24 

22.25 

19.75 

15.90 

0  .1095 

0  .1302 

0.165 

0  .2548 

3| 

0  .2956 

0  .06863 

26.61 

24.24 

21  .56 

17.42 

0  .0910 

0  .1095 

0.139 

0  .2122 

4 

0  .3356 

0  .08840 

28.44 

26.15 

23.29 

18.89 

0  .0796 

0  .0941 

0.119 

0.1806 

4* 

0  .3756 

0  .1168 

30.40 

27.98 

24.94 

20.39 

0  .0696 

0  .0823 

0.103 

0  .1565 

5 

0  .4204 

0  .1388 

32.48 

29.93 

26.72 

21  .80 

0  .0610 

0  .0703 

0.090 

0.1355 

6 

0  .5056 

0  .2008 

36.26 

33.45 

29.94 

24.54 

0  .0489 

0  .0575 

0  .0718 

0.1069 

7 

0  .5857 

0  .2694 

39.61 

36  .60 

33.05 

27.00 

0  .0410 

0  .0481 

0  .0598 

0  .0883 

8 

0  .6651 

0  .3474 

42.73 

39.53 

35.49 

29.30 

0  .0352 

0  .0412 

0.0511 

0  .0750 

9 

0  .7449 

0  .4356 

45.74 

42.25 

38.08 

31.52 

0  .0308 

0  .0351 

0  .0444 

0  .0648 

10 

0  .8348 

0  .5473 

48.97 

45.38 

40.85 

33.91 

0  .0268 

0  .0303 

0  .0377 

0.056 

11 

0  .9166 

0  .6599 

51.87 

48.06 

43.35 

36.08 

0  .0239 

0  .0279 

0  .0343 

0  .0495 

12 

1.0 

0  .7854 

54.70 

50.70 

45.77 

38.18 

0.0215 

0  .0250' 

0  .0304 

0  .0442 

13 

1  .1641 

0  .9531 

57.99 

53.84 

49.76 

40.66 

0  .0191 

0  .0222 

0  .0272 

0.039 

14 

1  .1875 

1  .1075 

60.70 

56.4 

50.21 

42.72 

0  .0174 

0  .0202 

0  .0248 

0  .0353 

15 

1  .2708 

1  .2675 

63.29 

58.78 

53.22 

44.66 

0  .0160 

0  .0186 

0  .0227 

0  .0323 

PRACTICAL   APPLICATIONS   OF  THE  FORMULA 


73 


TABLE  VII. 

CIRCULAR  CONDUITS. 

DIAMETERS,  INTERNAL  AREAS,  MEAN  HYDRAULIC  RADII  AND  THEIR  ROOTS. 


-,« 

3 

•a! 

.s 

§  ti 

J3  fe 

ll 

R 

Y^ 

V? 

If 

ll 

R 

Vr 

4^ 

"3 

c3   £_, 

1—1  "o> 

^  £ 

**  a 
II 

§| 

|  | 

|GC 

<& 

"fiCQ 

H-  1 

333 

1  .0408 

1    OfiO 

1  .0202 

^HJ 

366 

1   .UOU 

1  .0801 

'.0393 

ERR  A  TA 

)83 

1  .0993 

.0485 

5 

1  .118 

.057 

Page  72,  Table  VI  A,  for 
Loss   of    Head    in    Feet    per    Unit    Length    of 

533 

1  .137 
1  .415 
.173 

.066 
.0684 
.083 

Conduit  at  Unit  Velocity 

L66 

.1903 

1.091 

)83 

.2076 

1.099 

substitute 

55 

.226 

.275 

1  .107 
1.129 

Z,  or   Loss   of    Head   in  Feet  in  a   Length   of 

'5 

.323 
.369 

1.150 
1  .170 

R 

414 

.189 

2  g  Feet  at  Unit  Velocity. 

'5 

.457 

.208 

.5 

.224 

5 

.541 

.244 

" 

.581 

.257 

_ 

e» 

620 

273 

16 

1  .396 

0  .3333 

0  .5771 

0.759 

132 

95  03 

275 

^658 

l'288 

18 

1  .767 

0  .3750 

0  .6124 

0.782 

138 

103  '.87 

2.875 

.696 

1  .302 

20 

2.234 

0  .4166 

0.645 

0  .8031 

144 

113.10 

3.0 

1  .732 

1  .316 

22 

2.640 

0  .4583 

0  .6770 

0  .8220 

150 

122  .72 

3.125 

1  .768 

1  .330 

24 

3.142 

0.5 

0  .7071 

0.841 

156 

132  .76 

3.25 

1  .803 

1.343 

26 

3.687 

0  .5416 

0  .7360 

0.859 

162 

143  .16 

3  .375 

1  .837 

1.355 

28 

4.275 

0  .5833 

0  .7637 

0.874 

168 

153  .96 

3.5 

1  .871 

.368 

30 

4.909 

0.625 

0  .7906 

0.889 

174 

165.17 

3  .625 

1  .904 

.380 

32 

5.585 

0  .6666 

0  .8165 

0.904 

180 

176  .70 

3.75 

1  .937 

.392 

34 

6.305 

0  .7084 

0  .8416 

0.917 

186 

188  .70 

3.875 

1  .968 

.403 

•     36 

7.069 

0.75 

0.866 

0.931 

192 

201  .03 

4.0 

2.0 

.414 

38 

7.876 

0  .7916 

0  .8898 

0  .9435 

198 

213.8 

4.125 

2.031 

.425 

40 

8.927 

0  .8333 

0  .9129 

0  .9555 

204 

227.0 

4.25 

2.062 

.437 

42 

9.621 

0.875 

0  .9355 

0.967 

210 

240.5 

4.375 

2.091 

.446 

44 

10  .559 

0  .9166 

0  .9575 

0  .9784 

216 

254.5 

4.5 

2.121 

.456 

46 

11  .509 

0  .9584 

0  .9983 

0  .9991 

222 

268.8 

4.625 

2.151 

.466 

48 

12.566 

1  .0 

1.0 

1  .0 

228 

283.5 

4.75 

2.180 

.476 

50 

13  .635 

1  .0416 

1  .0206 

1  .0103 

240 

314.2 

5.0 

2.236 

.496 

72 


THE  FLOW  OF  WATER 


TABLE  VI.    A. 

WELDED   PIPES. 
TUBES    OF     BRASS,    GALVANIZED   IRON,    SHEET  IRON,   STEEL,   ETC. 


» 

1 

Actual 

Values  of  66  (  \/r  + 

^  /- 

w)VV 

Loss  of  Head  in  Feet  per 
Unit  Length  of  Conduit 

S| 

Diam- 

t>t  £»r 

Area  = 

eter 
in 

(P0.7854 

— 

1 

feet. 

n 

i 

0  .0225 

0  .00038 

6 

0  .0303 

0  .00072 

7 

1 

0.0411 

0  .00133 

g 

'l 

0  .0516 

0  .00209 

c 

1 

0  .0686 

0  .00370 

11 

1 

0  .0873 

0  .00599 

Ic 

a 

0.1150 

0  .01039 

1* 

a 

0  .1341 

0.01412 

1( 

2 

0.1722 

0  .02339 

J( 

2^ 

0  .2056 

0  .03320 

23 

3 

0  .2556 

0  .05130 

2' 

3^ 

0  .2956 

0  .06863 

2( 

4 

0  .3356 

0  .08840 

ft    1  1  R8 

2< 

5 

0  .4204 

U  .1  IDo 

0  .1388 

32  .48 

29.93 

tfX.    .£7-± 

26.72 

21  .80 

u  .uuyu 

0  .0610 

0  .0703 

XT"  .  J.T7O 

0.090 

0  .1355 

6 

0  .5056 

0  .2008 

36.26 

33.45 

29.94 

24.54 

0  .0489 

0  .0575 

0  .0718 

0.1069 

7 

0  .5857 

0  .2694 

39.61 

36  .60 

33.05 

27.00 

0  .0410 

0  .0481 

0  .0598 

0  .0883 

8 

0  .6651 

0  .3474 

42.73 

39.53 

35.49 

29.30 

0  .0352 

0  .0412 

0.0511 

0  .0750 

9 

0  .7449 

0  .4356 

45.74 

42.25 

38.08 

31  .52 

0  .0308 

0  .0351 

0  .0444 

0  .0648 

10 

0  .8348 

0  .5473 

48.97 

45.38 

40.85 

33.91 

0  .0268 

0  .0303 

0  .0377 

0.056 

11 

0  .9166 

0  .6599 

51.87 

48.06 

43.35 

36.08 

0  .0239 

0  .0279 

0  .0343 

0  .0495 

12 

1.0 

0  .7854 

54.70 

50.70 

45.77 

38.18 

0  .0215 

0  .0250" 

0  .0304 

0  .0442 

13 

1  .1641 

0  .9531 

57.99 

53.84 

49.76 

40.66 

0  .0191 

0  .0222 

0  .0272 

0.039 

14 

1  .1875 

1  .1075 

60.70 

56.4 

50.21 

42.72 

0  .0174 

0  .0202 

0  .0248 

0  .0353 

15 

1  .2708 

1  .2675 

63.29 

58.78 

53.22 

44.66 

0  .0160 

0  .0186 

0  .0227 

0  .0323 

PEACTICAL   APPLICATIONS   OF  THE   FORMULAE 


73 


TABLE  VII. 

CIRCULAR  CONDUITS. 

DIAMETERS,  INTERNAL  AREAS,  MEAN  HYDRAULIC  RADII  AND  THEIR  ROOTS. 


—  <  - 

.9 

I 

'cS  ° 

.S 

fi  S-. 

~^ 

CD  CD 

<£ 

R 

v7 

V? 

fl  (—  1 

|.s 

13  ® 

R 

Vr 

V' 

*c3  g 

g  c8 

1  1 

£  c3 

•g.s 

|I 

•i.i 

Jt 

i 

0  .001364 

0  .010417 

0  .10206 

0  .3195 

52 

14  .750 

1  .0833 

1  .0408 

1  .0202 

1 

0  .005454 

0  .02083 

0  .1444 

0.380 

54 

15  .904 

1.125 

1  .060 

1  .030 

H 

0  .012272 

0  .03125 

0  .1768 

0  .4302 

56 

17  .106 

1  .1666 

1  .0801 

1  .0393 

2 

0  .02182 

0  .04166 

0  .2039 

0  .4516 

58 

18  .347 

1  .2083 

1  .0993 

1  .0485 

2J 

0  .03163 

0  .0502 

0  .2240 

0  .4733 

60 

19  .635 

1  .25 

1  .118 

1.057 

3 

0  .04909 

0  .0625 

0.25 

0.5 

62 

20  .964 

1  .2966 

1  .137 

1  .066 

3i 

0  .06681 

0  .07292 

0.270 

0  .5196 

64 

22  .340 

1  .3333 

1  .415 

1  .0684 

4 

0  .08726 

0  .08333 

0.291 

0  .5370 

66 

23  .758 

1  .375 

1  .173 

1.083 

4^ 

0.11045 

0  .09375 

0  .3062 

0  .5533 

68 

25  .220 

.4166 

1  .1903 

1  .091 

5 

0  .1364 

0  .10416 

0  .3227 

0  .5681 

70 

26  .725 

.4583 

1  .2076 

1.099 

6 

0  .1963 

0.125 

0  .3535 

0.594 

72 

28.27 

.5 

.226 

1.107 

7 

0  .2672 

0  .1458 

0  .3819 

0  .6180 

78 

33.18 

.625 

.275 

1.129 

8 

0  .3490 

0  .1666 

0  .4082 

0.639 

84 

38.48 

.75 

.323 

1.150 

9 

0  .4418 

0  .1875 

0.433 

0.658 

90 

44.18 

.875 

.369 

1.170 

10 

0  .5585 

0  .2083 

0  .4564 

0.675 

96 

50.27 

2.0 

.414 

1.189 

11 

0  .6599 

0  .2297 

0  .4787 

0.692 

102 

56.75 

2.125 

1  .457 

1.208 

12 

0  .7854 

0.25 

0.5 

0  .7071 

108 

63  .62 

2.25 

1  .5 

1  .224 

13 

0  .9217 

0  .2708 

0  .5204 

0.721 

114 

70.88 

2.375 

1  .541 

1  .244 

14 

1  .0689 

0  .2916 

0.540 

0.735 

120 

78.54 

2.5 

1  .581 

1  .257 

15 

1  .2272 

0  .3125 

0  .5339 

0.748 

126 

86.59 

2.625 

1  .620 

1  .273 

16 

1.396 

0  .3333 

0  .5771 

0.759 

132 

95.03 

2.75 

.658 

1.288 

18 

1.767 

0  .3750 

0  .6124 

0.782 

138 

103  .87 

2.875 

.696 

1  .302 

20 

2.234 

0  .4166 

0.645 

0  .8031 

144 

113.10 

3.0 

.732 

1  .316 

22 

2.640 

0  .4583 

0  .6770 

0  .8220 

150 

122  .72 

3.125 

.768 

.330 

24 

3.142 

0.5 

0  .7071 

0.841 

156 

132  .76 

3.25 

.803 

.343 

26 

3.687 

0  .5416 

0  .7360 

0.859 

162 

143  .16 

3.375 

.837 

.355 

28 

4.275 

0  .5833 

0  .7637 

0.874 

168 

153  .96 

3.5 

.871 

.368 

30 

4.909 

0.625 

0  .7906 

0.889 

174 

165.17 

3.625 

.904 

.380 

32 

5.585 

0  .6666 

0  .8165 

0.904 

180 

176  .70 

3.75 

.937 

1  .392 

34 

6.305 

0  .7084 

0  .8416 

0.917 

186 

188  .70 

3.875 

.968 

1  .403 

36 

7.069 

0.75 

0.866 

0.931 

192 

201  .03 

4.0 

2.0 

1  .414 

38 

7.876 

0  .7916 

0  .8898 

0  .9435 

198 

213.8 

4.125 

2.031 

1.425 

40 

8.927 

0  .8333 

0  .9129 

0  .9555 

204 

227.0 

4.25 

2.062 

1.437 

42 

9.621 

0.875 

0  .9355 

0.967 

210 

240.5 

4.375 

2.091 

1  .446 

44 

10  .559 

0  .9166 

0  .9575 

0  .9784 

216 

254.5 

4.5 

2.121 

1  .456 

46 

11  .509 

0  .9584 

0  .9983 

0  .9991 

222 

268.8 

4.625 

2.151 

1  .466 

48 

12.566 

1.0 

1.0 

1  .0 

228 

283  .5 

4.75 

2.180 

1  .476 

50 

13  .635 

1  .0416 

1  .0206 

1  .0103 

240 

314.2 

5.0 

2.236 

1  .496 

74 


THE   FLOW   OF    WATER 


TABLE  VII.    A. 

ROOTS  OF  MEAN  HYDRAULIC  RADII. 


R 

v7 

V? 

R 

Vr 

^ 

R 

Vr 

v^ 

0.05 

0.224 

0.473 

2.75 

1.658 

1.287 

5.9 

2.429 

1.558 

0.1 

0.316 

0.562 

2.80 

1.673 

1.293 

6.0 

2.449 

1.565 

0.15 

0.387 

0.622 

2.85 

1  .688 

1.299 

6.1 

2.470 

1.571 

0.20 

0.447 

0.668 

2.90 

1.703 

1.305 

6.2 

2.490 

1  .578 

0.25 

0.5 

0.707 

2.95 

1  .718 

1  .311 

6.3 

2.510 

1  .584 

0.30 

0.548 

0.740 

3.0 

1  .732 

.316 

6.4 

2.530 

1  .590 

0.35 

0.592 

0.769 

3.05 

.746 

.322 

6.5 

2.550 

1  .597 

0.40 

0  .632 

0.803 

3.10 

.761 

.327 

6.7 

2.588 

1.609 

0.45 

0.671 

0.819 

3.15 

.775 

.332 

6.8 

2.608 

1.615 

0.5 

0.707 

0.841 

3.20 

.789 

.338 

6.9 

2.627 

1.621 

0.55 

0.742 

0.861 

3.25 

.803 

1.343 

7.0 

2.644 

1  .624 

0.60 

0.775 

0.881 

3.30 

.817 

1.347 

7.1 

2.665 

1  .630 

0.65 

0.806 

0.898 

3.35 

.830 

1.352 

7.2 

2.683 

1  .637 

0.70 

0.837 

0.914 

3.40 

.844 

1.358 

7.3 

2.702 

1  .643 

0.75 

0.866 

0.930 

3.45 

1.857 

1.363 

7.4 

2.720 

1.649 

0.80 

0.894 

0.946 

3.50 

1.871 

1.368 

7.5 

2.739 

1.655 

0.85 

0.922 

0.960 

3.55 

1.884 

1.373 

7.6 

2.757 

1.661 

0.90 

0.949 

0.974 

3.60 

1  .898 

1.378 

7.7 

2.775 

1.664 

0.95 

0.975 

0.987 

3.65 

1  .910 

1  .382 

7.8 

2.793 

1.670 

1.0 

1.0 

1.0 

3.70 

1.924 

1  .387 

7.9 

2.811 

1  .676 

1.05 

1.025 

1.012 

3.75 

1.936 

1  .392 

8.0 

2.828 

1  .682 

1.10 

1  .049 

1  .024 

3.80 

1.949 

.396 

8.1 

2.846 

1  .688 

1.15 

1.072 

1  .036 

3.85 

1.962 

.401 

8.2 

2.868 

1  .692 

1.20 

.095 

1.047 

3.90 

1.975 

.405 

8.3 

2.881 

1.697 

1.25 

.118 

1  .057 

3.95 

1  .987 

.410 

8.4 

2.898 

1.702 

1  .30 

.140 

1.068 

4.0 

2.0 

.414 

8.5 

2.915 

1  .707 

.35 

.162 

1  .079 

4.05 

2.012 

.419 

8.6 

2.933 

1.712 

.40 

.183 

1.088 

4.10 

2.025 

.423 

8.7 

2.950 

1.717 

.45 

.204 

1.097 

4.15 

2.037 

.427 

8.8 

2.966 

1.722 

.50 

.225 

.107 

4.20 

2.049 

1  .432 

8.9 

2.983 

1.727 

.55 

.245 

.115 

4.25 

2.062 

1  .436 

9.0 

3.0 

1.732 

.60 

.265 

.125 

4.30 

2.074 

1  .440 

9.1 

3.017 

1.737 

.65 

1  .285 

.133 

4.35 

2.086 

1  .444 

9.2 

3  .043 

1.741 

.70 

1  .304 

.142 

4.40 

2.098 

1.448 

9.3 

3.056 

1.746 

.75 

1  .323 

.150 

4.45 

2.111 

1  .453 

9.4 

3.066 

1  .750 

.80 

1  .342 

.158 

4.50 

2.121 

1  .457 

9.5 

3.082 

1.755 

.85 

1  .360 

.166 

4.55 

2.133 

1  .461 

9.6 

3.098 

1.760 

.90 

1  .378 

.174 

4.60 

2.145 

1  .466 

9.7 

3.114 

1.764 

.95 

1  .396 

.182 

4.65 

2.156 

.469 

9.8 

3.130 

.769 

2.0 

1  .414 

.189 

4.70 

2.168 

.473 

9.9 

3.146 

.773 

2.05 

1.432 

.196 

4.75 

2.179 

.476 

10.0 

3.162 

.777 

2.10 

1  .449 

1.204 

4.80 

2.191 

.480 

10.5 

3.240 

.800 

2.15 

1  .466 

1.211 

4.85 

2.202 

.484 

11.0 

3.317 

.820 

2.20 

1  .483 

1.218 

4.90 

2.214 

.488 

11.5 

3.391 

.845 

2.25 

1  .5 

1.225 

4.95 

2.225 

.492 

12.0 

3.464 

.860 

2.30 

1  .517 

1  .232 

5.0 

2.236 

.495 

12.5 

3.536 

.880 

2.35 

1  .533 

1  .238 

5.1 

2.258 

.503 

13.0 

3.606 

.899 

2.40 

.549 

.245 

5.2 

2.280 

.511 

13.5 

3.674 

.918 

2.45 

.565 

.251 

5.3 

2.302 

.518 

14.0 

3.742 

1  .934 

2.50 

.581 

.257 

5.4 

2.324 

.526 

14.5 

3.808 

1  .951 

2.55 

.597 

.263 

5.5 

2.346 

.532 

15.0 

3.873 

1  .968 

2.60 

.612 

.270 

5.6 

2.366 

.539 

15.5 

3.937 

1  .984 

2.65 

.628 

.276 

5.7 

2.387 

.548 

16.0 

4.0 

2.0 

2.70 

1  .643 

.282 

5.8 

2.408 

.552 

PRACTICAL   APPLICATIONS   OF  THE   FORMULAE  75 

TABLE  VIII. 

Table  VIII  contains  the  practically  most  useful  coefficients 
indicating  the  degree  of  roughness  of  a  conduit. 

In  the  design  of  a  new  conduit  it  is  well  to  remember,  that  the 
degree  of  roughness  of  a  conduit  is  not  a  permanent  quantity. 
Conduits  lined  with  cement,  smooth  concrete,  good  brickwork, 
planed  boards,  metals,  etc.,  gradually  deteriorate  and  assume  a 
degree  of  roughness  which  closely  resembles  that  of  a  sawed 
board  (m  =  0.68) ,  in  case  of  sewers  that  of  common  brick  work 
(m  =  0.57) ,  in  case  of  large  riveted  pipes  that  of  very  rough 
brick  work  (m  =  0.45) . 

If  the  velocity  is  feeble,  or  the  flow  often  interrupted,  crypto- 
gamic  plants  sooner  or  later  appear  on  the  walls  of  open  conduits 
and  rust  or  calcareous  matter  coates  the  walls  of  pipes.  In 
such  a  condition  the  degree  of  roughness  corresponds  to  that  of 
very  rough  brick  work  (m  =  0.45) . 

If  left  to  themselves,  channels  in  earth  of  all  descriptions 
likewise  deteriorate  and  gradually  assume  a  degree  of  roughness 
corresponding  to  that  of  a  natural  channel  (k  =  1.93  in  most 
cases) . 

For  artificial  channels  in  earth  Table  VIII  gives  values  of 
both  m  and  k.  Owing  to  the  abnormally  rapid  decreases  in  the 
value  of  c  with  the  decrease  of  the  depth  of  the  water  in  rough 
channels  in  earth  a  negative  value  of  m  gives  better  results 
than  k. 

The  k  formula,  however,  gives  good  results  in  all  cases  where  R 
is  greater  than  one  foot. 

The  relation  between  K  and  m  and  the  coefficient  n  of  the  for- 
mula of  Kutter  is  given  by 

1  +  K        0.02 
100      =  1  +  m  ' 


76  THE   FLOW  OF   WATER 


TABLE   VIII. 

VALUES  OF  m  AND  k,  THE  COEFFICIENTS  INDICATING  THE  DEGREE  OF  ROUGH- 
NESS.    A.     CONDUITS  UNDER  PRESSURE. 


1.0 
0.95 


0.83 


0.68 


0.57 
0.53 
0.45 


0.30 
0.20 


Description  of  Conduit. 


New,  straight  tin  or  plated  pipes. 

Pipes  of  planed  boards  or  clean  cement,  new.  Very  smooth  new 
asphalt-coated  cast  and  wrought-iron  pipes.  New  asphalt-coated 
riveted  pipes  not  exceeding  6  inches  in  diameter. 

Ordinary  new  asphalt-coated  cast  and  wrought-iron  pipes.  Wrought- 
iron  pipes  not  coated,  new.  Glass  and  lead  pipes.  Pipes  lined  with 
smooth  concrete  or  cement  plaster. 

Pipes  lined  with  cement  or  smooth  concrete,  pipes  of  planed  or  rough 
boards,  cast  and  wrought-iron  pipes,  coated  or  not  coated,  steel 
and  wrought-iron  riveted  pipes  not  exceeding  3  feet  in  diameter 
(all  some  time  in  use  but  fairly  clean). 

Sewer  pipe.   Conduits  lined  with  common  brickwork  or  rough  concrete. 

New  asphalt-coated  steel-riveted  pipe  exceeding  3  feet  in  diameter. 

Conduits  lined  with  very  rough  brickwork  or  very  rough  concrete. 

Steel-riveted  pipe  exceeding  3   feet  in  diameter,   some  years  in  use. 

Old  cast  and  wrought-iron  pipes  of  all  descriptions,   not  very  clean. 

Old  steel-riveted  pipe  exceeding  3  feet  in  diameter. 

Drain  tile. 


B.     OPEN  CONDUITS. 


Description  of  Conduit. 


1  .0 

0.95 

0.83 

0.80 
0.70 


0.57 


0.45 

0.30 
0.15 
0.0 


Conduits  lined  with  neat  cement  exceptionally  smooth. 

New  conduits  lined  with  neat  cement  or  planed  boards. 

New  brick  conduits  washed  with  cement,  conduits  smoothly  dressed 
with  cement  mortar. 

New  conduits  lined  with  smooth  concrete  or  very  good  brick  work. 

Conduits  lined  with  sawed  boards  or  fairly  good  brick  work. 

Aqueducts  lined  with  neat  cement,  cement  plaster,  smooth  concrete 
very  good  brickwork,  planed  boards  (all  some  time  in  use). 

Channels  lined  with  common  brickwork,  rough  concrete  or  smoothly 
dressed  ashlar  masonry.  Sewers  lined  with  neat  cement,  smooth 
concrete,  brickwork  washed  with  cement  or  plastered  with  cement 
mortar,  fairly  good  and  very  good  brickwork  (all  some  time  in  use.) 

Channels  lined  with  very  rough  brickwork  or  concrete,  fairly  good 
ashlar  masonry. 

Channels  lined  with  common  ashlar  or  very  good  rubble  masonry. 

Channels  lined  with  roughly  hammered  stone  masonry. 

Channels  lined  with  common  rubble  masonry.     Channels  in  rockwork. 


PRACTICAL  APPLICATIONS   OF   THE   FORMULAE 


77 


TABLE  VIII.  —  Continued. 
C.     CHANNELS  IN  EARTH. 


m 

k 

Description  of  Conduit. 

0.57 
0.15 
0.0 

-0.10 
-0.20 

-0.32 

0.27 
0.74 
1.0 

1.2 
1  .5 

1.93 

Channels  of  very  regular  cross-section  in  stiff  clay  or  clayey 
loam. 
Channels  of  fairly  regular  cross-section   in   fine   cemented 
gravel. 
Channels  of  fairly  regular  cross-section  in  coarse  cemented 
gravel. 
Channels  in  rock  work. 
Fairly  regular  channels  in  sand  or  sand  with  gravel  imbedded. 
Fairly  regular  channels  in  earth,  tolerably  free  from  stones 
and  plants. 
Ordinary  channels  in  earth  or  gravel.     Channels  with  stones 
vegetation  or  other  impediments  to  flow. 
Natural  channels,  creeks,  rivers. 

TABLE  IX. 

ALPHABETICAL  LIST  OF  AUTHORITIES  WHOSE  EXPERIMENTAL  DATA  ARE 
GIVEN  IN  TABLE  X,  METHODS  OF  GAUGING  AND  PUBLICATION  CONTAINING 
ORIGINAL  RECORD  OF  EXPERIMENTS. 


Author. 

Description  of 
Channel  Gauged. 

Method  of  Gauging. 

Where  Recorded. 

Adams,  A.  L. 

Wooden         Stave 
pipes. 

Discharge  meas- 
ured by  rise  in 
reservoir  surface, 
loss  of  head  by 
open  standpipes. 

Engineering  News. 
Sept.  1898. 

Baumgarten    .    . 

Aqueduct.       Bot- 
tom   of    cement, 
sides     of     brick. 

Piezometer. 

Darcy-Bazin. 
Recherches    hy- 
drauliques. 

Benzenberg     .    . 

Brick    sewer 

Floats    probably 

Trans  A    S  C    E 

Bossut  .... 

Tin       and     Lead 
pipes. 

Discharge  meas- 
ured in  tanks. 

Hamilton     Smith, 
Hydraulics,  1886. 

Bruce   

Aqueduct.       Con- 
crete. 

Discharge  meas- 
ured by  rise  in 
reservoir  surface. 

Proceedings       of 
Institute  of  C.  E. 
London,  1896. 

Brush  

Cast-iron  pipes. 

Quantities  meas- 
ured at  pumps. 

Quoted  by  Kutter. 
"The     Flow     of 
Water." 

Clarke  

Brick  sewer. 

Discharge  meas- 
ured by  rise  in 
reservoir  surface. 

H.  Smith.       Hy- 
draulics, 1886. 

78 


THE   FLOW   OF   WATER 


TABLE  IX. — Continued. 


Author. 

Description  of 
Channel  Gauged. 

Method 
of  Gauging. 

Where  Recorded. 

Cunningham  .    . 

Aqueduct            of 
masonry.     Ganges 
Canal. 

Velocities  meas- 
ured by  one-inch 
tin  rod  floats. 

Roorkee  Hydr.  Ex- 
periments,   1880. 

Darcy  

Pipes. 

Discharge  meas- 
ured in  tanks,  loss 
of  head  by  piezo- 
meter or  mercury 
column. 

Experiments     sur 
le  mouvement  de 
Peau     dans    les 
tuyaux. 

Darcy-Bazin   .    . 

Cement  and    con- 
crete    conduits. 

Discharges  meas- 
ured by  orifices 
previously  tested. 

Recherches    hydr. 
Paris   1865. 

Darcy-Bazin  .    . 

Conduits  of  planed 
and  rough  boards, 
or  lined  with  brick. 

Discharges    meas- 
ured   by   orifices 
previously  tested, 
20    centimeter 
square. 

Recherches    hydr. 
Paris,  1865. 

Darcy-Bazin  .    . 

Canal    lined    with 
ashlar    masonry. 

Piezometer. 

Recherches    hydr. 
Paris,  1865. 

Darcy-Bazin  .    . 

Tailrace  lined  with 
ashlar     masonry. 

Discharge  meas- 
ured by  orifice 
50  centimeter 
square. 

Recherches    hydr. 
Paris,  1865. 

Darcy-Bazin  .    . 

Tunnel  lined  with 
ashlar    masonry. 

Current  meter 
and  reservoir. 

Recherches  hydr. 
Paris,  1865. 

Darcy-Bazin  .    . 

Section    of    Gros- 
bois    canal  lined 
with  masonry. 

Current  meter 
and  reservoir. 

Recherches    hydr.  • 
Paris,  1865. 

Darcy-Bazin 

Chazilly       Canal. 

Piezometer,  cur- 
rent meter  and 
reservoir. 

Recherches  hydr. 
Paris,  1865. 

Darcy-Bazin 

Grosbois    Canal. 

Piezometer,  cur- 
rent meter  and 
reservoir. 

Elecherches    hydr. 
Paris,  1865. 

Dubuat 

Canal  du  Jard. 

Surface  floats. 

Principes    hydr. 
Paris,  1786. 

Ehman     .... 

Galvanized       and 
cast-iron    pipes. 

Discharges  meas- 
ured by  volumes. 

Iben    Druckho- 
henverlust. 

Fanning  .... 

Cement  lined  pipe. 

Weir        measure- 
ment  probably. 

Water         Supply 
Engineering. 

PRACTICAL  APPLICATIONS  OF  THE  FORMULAE 


79 


TABLE  IX. — Continued. 


Author. 

Description  of 
Channel  Gauged. 

Method 
of  Gauging. 

Where  Recorded. 

Fteley  &  Stearns 

Sudbury    Conduit. 
Brick  coated  with 
cement   and   not 
coated. 

Weir      measure- 
ment. 

H.  Smith. 
Hydraulics,  1886. 

Fortier    .... 

Irrigation  Chan- 
nels. 

Current      meter. 

U.    S.    Geol.  Sur- 
vey.    Irr.  Papers, 
1901. 

Hawks     .... 

Steel-riveted  pipe. 

Weir  measure- 
ment. 

Tr.  A.  Soc.  C.  E., 
1899. 

Herschel      .    .    . 

Steel-riveted  pipes. 

Discharges  meas- 
ured by  Ventury 
meter,  loss  of 
head  by  Bourdon 
gauges. 

115    Experiments. 

Horton    .... 

Brick  sewers 
washed  with  ce- 
ment. 

Weir      measure- 
ment probably. 

Eng.  News. 

Hubbel  &  Fenkell 

Cast-iron       ppes. 

Tr.  A.  Soc   C   E 

1898. 

Iben 

Oast-iron  pipes. 
Pipes  coated  with 
tar. 

Discharges  meas- 
ured by  volumes, 
loss      of      head 
by            pressure 
gauges. 

Druckhohenver- 
lust. 

Kuichling    .    .    . 

Riveted     pipes. 
Cast-iron     pipes. 

Quantities  meas- 
ured by  rise  in 
reservoir  surface, 
loss  of  head  by 
mercury  gauges. 

Marx-Wing          & 
Hoskins,  Tr.  A.  S. 
C.  E.,  1898-1899. 

Kutter     .... 

Channels  lined 
with  rubble  ma- 
sonry. 

Surface  floats. 

)ie  neue  Theorie. 

La  Nicca     .    .    . 

Alpine      Streams. 

Surface    floats. 

Kutter,  Die  neue 
Theorie. 

Lampe     .... 

3ast-iron       pipes. 

Discharges  meas- 
ured by  reservoir 
contents,  loss  of 
head  by  pressure 
gauges. 

ben. 
)ruckhohenver-  ' 
lust. 

Legler      .        .    . 

Canals. 

Rodfloats. 

lydrotechnische 
Mittheilungen. 

80 


THE  FLOW   OF   WATER 


TABLE   IX.  —  Continued. 


Author. 

Description  of 
Channel  Gauged. 

Method 
of  Gauging. 

Where  Recorded. 

McDougall      .     . 

Irrigation      chan- 
nels. 

Current    meter. 

U.  S.  Geolog.  Sur- 
vey Irr.  Papers. 

Marx-Wing        & 
Hoskins     .    .    . 

Riveted    pipe. 
Stave   pipe. 

Discharges    meas- 
ured by  Venturi 
meters,     loss     of 
head  by  mercury 
gauges. 

Trans.  A.  S.  C.  E., 
1898. 

Noble               .    . 

Stave  pipes. 

Trans.  A  S    C  E  , 

1902. 

Passini  &  Gioppi 

Aqueduct.        Bot- 
tom of   concrete, 
sides  of  brick. 

Current  meter. 

Giornale          del. 
Genio  Civile.  Ro- 
ma, 1893. 

Passini  &  Gioppi 

Syphon    aqueduct 
of  brick. 

" 

tt 

Passini  &  Gioppi 

Canal    Cavour. 

" 

11 

Perrone    .... 

Aqueduct     coated 
with  clean  cement. 

" 

Zoppi:     Sul    Vol- 
turno,    Carte  hy- 

Perrone    .... 

Tunnel    in     rock- 
work. 

" 

Italic. 

Rafter  

Riveted  pipes 

Discharge     meas- 

Tr. A.  S.  C.  Eng  , 

ured    by   rise   in 
reservoir  surface, 
loss    of   head  by 
piezo-meter. 

XXVI. 

Revy    .    .    .    .    . 

La  Plata  and  Pa- 
rana Rivers. 

Current  meter. 

Hydraulics          of 
great  rivers,  Lon- 
don, 1881. 

Rittinger     .    .    . 

Channels        lined 
with           rubble 
masonry. 

Discharges  meas- 
ured    in     tanks. 

Bornemann  :     Der 
Civil    Ingenieur. 

Roff  

Saalach   River. 

Piezometer. 

Grebenau,  Theorie 

der  Bewegung  des 
Wassers. 

Rowland      .    .    . 

Wrought-iron 
pipes. 

Discharge     meas- 
ured by  volumes. 

Brush. 

Smith,  H     .    .    . 

Riveted  pipes 

Velocities      meas- 

Tr A  S  C  E 

ured     by     weirs 
and         Standard 
orifices. 

"Hydraulics," 
N.  Y.,  1886. 

PRACTICAL  APPLICATIONS   OF  THE   FORMULAE 


81 


TABLE  IX.  —  Concluded. 


Author. 

Description  of 
Channel  Gauged. 

Method 
of  Gauging. 

Where  Recorded. 

Smith,  J.  W.  .    . 

Riveted       pipes. 

Discharge     meas- 
ured    by     weir, 
loss  of  head   by 
piezometers. 

Tr.  A.  S.  C.  E 
Vol.  XXVI. 

Stearns     .... 

Brick   aqueduct. 

Current  meter. 

Report     to     the 
New  Croton  Aque- 
duct   Com.,  1895. 

Schwartz     .    .    . 

Weser  River. 

Current  meter. 

Funk:    Beit  rag  zur 
allgemeinen 
Wasserbaukunst 
Lemgo,  1808. 

Wampfler   .    .    . 

3anal. 

Surface  floats. 

Kutter:    Die  neue 
Theorie. 

TABLE  X. 

Table  X  contains  the  most  reliable  experimental  data  from 
which  the  general  formula  is  deduced.  For  conduits  under 
pressure  the  numerical  values  of  (a),  the  coefficient  of  variation 
of  c  is  generally  given.  This  is  done  in  order  to  show  the  details 
of  the  variation  of  c.  For  open  conduits  the  variation  is  generally 
indicated  by  giving  the  velocity  roots.  These  roots  are  found 
from  the  formula 

=  log.  vt — log.  % 

log  (66  %/r  +  m)  Vrs)^  log.  (66  ( t/r  +  m)Vrs~)Q 

The  value  of  x  being  thus  found  the  value  of  m  is  found  by 
putting 

—  -\Jr  =  m 


or 


66  a 


=  m 


For  artificial  channels  in  earth  the  values  of  m  have  been 
given  in  addition  to  the  values  of  K.  Owing  to  the  abnormally 
rapid  decrease  in  the  values  of  c  with  the  decrease  of  the  depth 
of  the  water  in  very  rough  channels  a  negative  value  of  m  gives 
better  results  than  K.  The  K  formula,  however,  gives  good 
results  in  all  cases  where  R  is  greater  than  one  foot. 


82 


THE  FLOW   OF  WATER 


TABLE  X. 

EXPERIMENTAL  DATA. 
I.   RIVETED  PIPES. 


Description  of  Conduit. 

m 

L 

d 

R 

1000s 

V 

c 

a 

0.80 
.04 
.130 
.219 

.287 
.291 

New  straight  asphalt- 
coated  wrought-iron 
riveted    pipe    with 
screw  joints. 
—  Darcy. 

0.94 

365 

0.271 

0  .0677 

0.27 

2.028 
12.20 
40.70 
106  .54 
156  .05 

0.328 
1.171 
3.117 
6.148 
10  .535 
12  .786 

76.7 
99.9 
108.4 
117.1 
124.0 
124.3 

Do. 

0.92 

365 

0.643 

0  .1607 

0.20 
1  .29 
5.80 
12.0 
29.7 
121  .56 

0.591 
1  .529 
3.53 
5.509 
9.00 
19.72 

104.1 
106.2 
115.6 
125.4 
130.2 
141  .0 

1.013 
1  .035 
1  .125 
1  .220 
1.267 
1.372 

... 

... 

Do. 

0.82 

365 

0.935 

0.234 

0.70 
4.33 
11  .90 
28.07 

1  .296 
3.868 
6.673 
10  .522 

101.3 
121.6 
126.5 
129.9 

1.013 
1  .216 
1  .265 
1  .299 

Sheet-iron         riveted 
pipe     with    funnel 
mouthpiece    7.8  ft. 
long. 
—  Hamilton  Smith. 

0.68 

700 

0.911 

0.228 

8.50 
13.34 
16.95 
25.59 
33.09 

4.712 
6.094 
6.927 
8.659 
10  .021 

107.1 
110.6 
111  .5 
113.4 
115.5 

1.19 
1.229 
1.240 
1  .260 
1  .283 

Do. 
Coated  with  asphalt. 
Funnel  mouthpiece 
12  ft.  long. 

0.68 

700 

1.056 

0.264 

6.68 
14.28 
22.19 
33.18 

4.595 
6.962 
8.646 
10  .706 

109.4 
113.4 
113.0 
114.4 

.200 
.242 
.237 
.253 

Do. 
Funnel      mouthpiece 
14.8  ft.  long. 

0.69 

700 

1.229 

0.307 

5.02 
10.97 
12.27 
16.46 
24.70 
32.31 

4.383 
6.841 
7.314 
8.462 
10  .593 
12.09 

111  .6 
119.8 
119.1 
119.2 
121  .6 
121.3 

.181 
.246 
.261 
.260 

.286 

.285 

Do. 
Double    riveted   pipe 
with     some     easy 
curves. 

0.65 

4440 

1.416 

0.354 

66.72 

20  .143 

131  .1 

1.395 

Do. 

0.69 

1200 

2.154 

0.538 

16.41 

12  .605 

134.1 

1  .325 

Do. 
Inverted  syphon  with 
887  ft.  depression. 

0.63 

12800 

2.43 

0.607 

11.72 

10.78 

127.8 

1.30 

DESCRIPTION  OF  CONDUIT,   ETC. 


83 


TABLE  X.  —  Continued. 


Description  of  Conduit. 

m 

L 

d 

R 

1000s 

V 

c 

a 

Wrought-iron   riveted 
pipe  with  lap  joints. 

Paint     coating     worn 
off,  somewhat  rusty. 
—  Clemens-Herschel. 

0.54 

152.9 

8.58 

2.145 

0  .0079 
0.032 
0.0837 
0  .1557 
0  .2453 
0.354 
0  .4991 
0  .6619 
0  .8470 

0.50 
1.0 
1  .5 
2.0 
2.5 
3.0 
3.5 
4.0 
4.5 

126.9 
116.6 
111.9 
109.4 
109.0 
108.2 
107.0 
106.2 
105.6 

1  .089 
1.00 
0.959 
0.938 
0.934 
0.928 
0.917 
0.910 
0.905 

Asphalt-  coated   steel- 
riveted   pipe.  —  A. 
McL.  Hawks. 

0.55 

1.166 

0  .2915 

0  .4550 
0.584 

0.932 
1  .136 

82.2 
86.0 

0.98 
1.026 

Asphalt  coated  cylin- 
der      joint      steel 
pipe.  —  A.L.Adams. 

0.65 

16416 

1.33 

0.333 

5.0 

4.58 

110.0 

1.183 

Asphalt-  coated     cylin- 
der joint  steel-  riveted 
pipe     with     curves. 
-E.   Kuichling. 

0.57 

91641 

3.166 

0  .7915 

1.01 
0.99 

3.23 
3.27 

114.0 
116.6 

1.14 
1.166 

Asphalt-coated    taper 
joint     steel-  riveted 
pipe.   New.  —  Clem- 
ens-Herschel. 

0.56 

81139 

3.5 

0.875 

0.112 

1  .0 
2.0 
3.0 
4.0 
5.0 
6.0 

101  .0 
104.3 
106.4 
107.8 
108.4 
108.5 

1.0 
1.032 
1.053 
1.067 
1.073 
1.074 

Do. 

0.54 

5574 

3.5 

0.875 

0.13 

1.0 
2.0 
3.0 
3.5 
4.0 
5.0 
6.0 

96.0 
107.9 
112.6 
113.0 
112.8 
110.8 
110.0 

0.96 
1.08 
1.128 
1.132 
1.130 
1.111 
1.102 

Do. 
Cylinder  joint,  many 
curves. 

0.53 

24000 

4.0 

1  .0 

0  .0976 

1.0 
2.0 
3.0 
3.5 
4.0 
5.0 
6.0 

101  .2 
108.3 
112.8 
113.4 
113.2 
112.0 
111.6 

1.0 

1  07 
1  113 
1  .119 
1  .118 
1.105 
1.091 

Asphalt-coated  cylin- 
der      joint      steel- 
riveted  pipe.  —  J.  W. 
Smith. 

0.80 
0.74 

39809 
34176 

2.916 
2.75 

0.584 
0.55 

1.31 
1.31 

3.52 
3.96 

126.8 
123.2 

1.15 
1.166 

84 


THE  FLOW   OF   WATER 


TABLE  X.  —  Continued. 


Description  of  Conduit. 

m 

L 

d 

R 

1000  s 

V 

c 

a 

Asphalt-coated    butt- 
jointed  steel-riveted 
pipe     with     many 
curves.  —  Marx-  Wing. 

0.50 

4367 

6.00 

1.5 

0.07 
0.16 
0.24 
0.559 
0.495 
0.776 

1  .08 
1.57 
2.14 
2.59 
3.02 
3.84 

108.0 
114.0 
113.0 
110.0 
112.0 
113.0 

i.o 

1.055 
1  .046 
1  .018 
1  .037 
1  .046 

II.     OLD  RIVETED  PIPES. 


Cylinder  joint  asphalt- 
coated  steel-  riveted 
pipe.  Fourteen 
years  in  use. 
—George  W.  Rafter. 

0.31 

45400 

2.0 

0.50 

3.83 
3.58 
3.46 

3.32 
3.32 
3.35 

76.0 
78.0 
80.5 

1  .0 
1.030 
1.06 

Do. 

0.29 

45400 

3.0 

0.75 

0.45 
0.43 

1.47 
1.49 

80.4 
83.0 

1.0 

Do. 
One  year  in  use.  —  E. 
Kuichling 

0.59 

45400 

3.17 

0  .7915 

1  .59 
1.61 
1  .62 

3.88 
3.91 
3.90 

109.3 
109.3 
109.1 

1  .08 
1  .08 
1.08 

Taper  joint,  steel 
riveted.  Four  years 
old  .  —  Clemens-Her- 
schel. 

0.26 

24648 

4.0 

1.0 

... 

1  .0 
1.5 
2.0 
2.5 
3.0 
3.5 
4.0 
5.0 
6.0 

78.0 
84.6 
89.6 
92.4 
93.0 
93.2 
94.2 
94.4 
94.9 

0.94 
1  .019 
1  .080 
1  .113 
1.120 
1.121 
1.135 
1  .137 
1  .143 

... 

Cylinder  joint  steel- 
riveted  pipe,  four 
years  in  use.  —  C. 
Herschel. 

0.47 

24600 

4 

1  .0 

1.0 
1  .5 
2.0 
2.5 
3.0 
3.5 
4.0 
5.0 
6.0 

97.2 
100.8 
103  .3 
104  .9 
105.3 
104.8 
104.0 
103.9 
103.7 

1.0 
1.024 
1.062 
1  .079 
1  .083 
1  .079 
1.069 
1.066 
1  .066 

III.   NEW  WROUGHT-IRON  PIPES,  NOT  COATED. 

Straight           pipe.  — 
Darcy. 

0.83 

372 

0  .0873 

0  .0218 

0.33 
10.15 
43.48 
105  .71 
309  .52 

0.19 
1.207 
2.612 
4.203 
7.166 

70.7 
81.1 

84.8 
87.5 
87.2 

0.877 
1  .006 
1  .052 
1.098 
1  .094 

... 

DESCRIPTION   OF  CONDUIT,    ETC. 


85 


TABLE  X.  —  Continued. 


Description  of 
Conduit. 

m 

L 

d 

R 

1000  s 

V 

c 

a 

Straight  pipe. 
—  Darcy. 

0.83 

372 

0  .1296 

0  .0324 

0.22 
3.36 
23.89 
123  .15 
224  .08 

0.205 

0.858 
2.585 
6.300 
8.521 

76.9 

82.3 
92.9 
99.8 
100.0 

0.824 
0.989 
1  .128 
1  .212 
1  .215 

Do. 

Rowland. 

0.83 

31  .0 
31.0 
31.0 
97.0 

0  .0833 
0  .0833 

0  .0208 
0  .0208 

6258  .6 
8935  .5 
10741  .9 
2000  .0 
2855  .6 
3432  .9 

36.1 
43.4 
48.1 
19.9 
24.5 
27.2 

100.0 
100.6 
101.7 
97.5 
100.5 
101  .7 

1  .240 
1.248 
1  .261 
1  .209 
1.247 
1  .261 

IV.   PIPES  COATED  WITH  TAR. 


New  cast-iron    pipe. 
—  Iben           1876 

0.66 

415 

0.335 

0.084 

1.98 
4.11 

1.0 
1  .70 

79.0 
92.0 

1.0 
1  164 

Quoted  by  Kutter 

6.56 

2.10 

90  0 

1   139 

7.83 

2.30 

90  .0 

1   139 

11  .07 

2.80 

91  .0 

1  .152 

Do. 

0.56 

1093 

0.50 

0.125 

4.59 
11  .62 

2.00 
3  30 

82.0 
87  0 

1  .080 
1   144 

16.21 

3.90 

88  0 

1   156 

22.32 

4.80 

92  0 

1  210 

30.27 

5.30 

87.0 

1  .144 

Do. 

0.65 

1795 

1  .001 

0.25 

1.46 
1  830 

1.60 
2  10 

85.0 
97  0 

0.944 
1  080 

2.19 

2.60 

112  0 

1  244 

3  .84 

3  .80 

121  0 

1  .344 

6.03 

4.80 

125.0 

1  .388 

Do. 

0.69 

3514 

1.667 

0.417 

0.12 
0.48 

0.70 
1  .60 

105.0 
110.0 

1.077 
1  .125 

0.76 

1  .90 

109  0 

1  115 

1.21 

2.50 

109.0 

1  .115 

V.     ASPHALT-COATED,    WROUGHT   AND    CAST-IRON    PlPE.       NEW. 


Asphalt-coated 
wrought-iron  pipe 
with  funnel  mouth- 
piece. —  Hamilton 
Smith. 

0.89 

60 

0  .0875 

0  .0218 

26.93 
52.19 
103  .38 
130  .64 

2.22 
3.224 
4.761 
5.443 

91.6 
95.5 
100.2 
101  .9 

1.095 
1  .140 
1.189 
1.205 

... 

Asphalt-coated  cast- 
iron  pipe.  —  Darcy. 

0.90 

366 

0  .4495 

0.1124 

0.24 
4.25 
22.25 
98.52 
167  .56 

0.489 
2.503 
5.623 
11  .942 
15  .397 

94.1 
108.4 
112.5 
113.5 
112.2 

0.960 
1  .107 
1.150 
1.160 
1.150 

86 


THE  FLOW   OF   WATER 


TABLE  X.  —  Continued. 


Description 
of  Conduit. 

m 

L 

d 

R 

1000s 

V 

c 

a 

Asphalt-coated   pipe 
five  years  old,  in 
good        condition. 
—  Lampe. 

0.83 

26000 

1.373 

0.343 

0.594 
1.376 
1.63 
1.95 

1.577 
2.479 
2.709 
3.090 

110.5 
114.1 
114.6 
119.4 

1.072 
1  .107 
1  .112 
1.162 

Cast-iron  pipe.  — 
Darcy. 

0.81 

365 

0  .6168 

0  .1542 

3.68 
22.50 
109  .80 
145  .91 

2.487 
6.342 
14  .183 
16.168 

104.4 
107.7 
109.0 
107.8 

1.107 
1.142 
1.155 
1  .142 

Asphalt-coated  pipe, 
four  years  in  use. 
—  Ehmann. 

0.80 

810 

0.662 

0.166 

0.367 
0.850 
1  .332 
1.883 

0.73 
1.12 

1.45 
1  .69 

92.7 
94.7 
97.9 
96.0 

0.984 
1.01 
1  .039 
1.017 

Asphalt-coated  cast- 
iron  pipe.  —  Hu  li- 
bel and  Fenkel. 

0.83 

... 

1.0 

0.25 

... 

1.0 
2.0 
3.0 
4.0 
5.0 

101  .5 
109.6 
114.6 
118.3 
121  .5 

1  .0 
1  .08 
1.13 
1.166 
1  .196 

Cast-iron       pipe.  — 
Darcy. 

0.77 

365 

1  .6404 

0.41 

0.45 
1  .20 
2.10 
2.60 

1  .472 
2.602 
3.416 
3  .674 

108.4 
117.3 
116.4 
112.5 

1.047 
1  .135 
1.126 
1.090 

Cast-Iron  Force 
Main.  Large  num- 
ber of  summits, 
angles  and  curves, 
amongst  which 
there  are  four 
right  angles  and 
ten  quadrants  of 
30  ft.  radius.  — 
Brush. 

0.80 

75000 

1.667 

0.417 

0.733 
0.880 
.026 
1.187 
.333 
.493 
1  .64 
1.800 

2.0 
2.24 
2.36 
2.52 
2.68 
2.76 
2.92 
3.0 

114.4 
117.0 
.114  .1 
113.3 
113.7 
110.6 
111  .7 
109.5 

1  .08 
1  .105 
1  .080 
1  .071 
1.071 
1.045 
1  .055 
1  .035 

.  .. 

Asphalt-  coated  cast- 
iron  pipe.  Some 
easy  vertical  cur- 
ves. —  Stearns. 

0.97 

1747 

4.0 

1.0 

0.318 
0.711 
1  .221 
1  .849 

2.616 
3.738 
4.965 
6.195 

146.7 
140.1 
142.1 
144.1 

FA 

1.077 
1.093 
1  .109 

... 

VI.  OLD  CAST  AND  WROUGHT-!RON  PIPES. 

Old  cast-iron    pipe. 
—  Darcy. 

0.52 

366 

0  .2628 

0  .0657 

0.84 
7.25 
16.10 
45.35 

0.458 
1.463 
2.224 
3.777 

62.0 
67.3 
68.7 
68.9 

0.94 
1.04 
1.062 
1.065 

DESCRIPTION   OF  CONDUIT,    ETC. 


87 


TABLE  X.  —  Continued. 


Description  of 
Conduit. 

m 

L 

d 

R 

1000s 

V 

c 

a 

Old    cast-iron    pipe, 
cleaned.  —  Darcy. 

0.85 

0.84 
7.23 
15.57 
44.73 

0.633 
2.014 
2.835 
5.007 

85.2 
92.4 
88.6 
93.4 

1  .00 

1  .08 
1  .026 
1  .092 

Old   cast-iron    pipe. 
—  Darcy. 

0.45 

365 

0.798 

0  .1995 

0.94 
4.73 
22.90 
41  .05 
139  .81 

1  .007 
2.32 
5.095 
6.801 
12  .576 

73.6 
75.5 
75.1 
75.2 
75.3 

1  .0 
1.023 
1.02 
1  .02 
1  .02 

Old    cast-iron    pipe, 
twelve  years  in  use. 
Slightly    tubercu- 
lated.  —  I  ben. 

0.45 

541 

1.0 

0.25 

2.24 

2.84 

1.79 
2.03 

75.7 
76.2 

1  .0 
1.01 

Do.,  two  years  in  use, 
slightly  incrusted. 

0.56 

2149 

1.0 

0.25 

0.26 
0.41 
0.81 
1.28 
2.99 

0.60 
0.80 
1.20 
1.60 
2.40 

74.5 
81.0 
85.0 
92.0 
86.0 

0.878 
0.962 
1  .01 
1  .092 
1  .021 

Do.,  fourteen  years  in 
use,     slightly    in- 
crusted. 

0.39 

7179 

1.00 

0.25 

0.42 
1  .65 
4.44 
9.43 

0.70 
1  .60 
2.70 
3.90 

71.0 

78.0 
80.0 
80.0 

0.979 
1  .075 
1  .133 
1.133 

Do.,  fifteen  years  in 
use.     Heavily  in- 
crustated. 

0.30 

1808 

1.00 

0.25 

0.65 
3.76 
6.12 
7.13 

0.90 
1  .80 
2.30 
2.60 

67 

58 
59 

58 

0.991 
0.860 
0.875 
0.860 

Do.,  twenty  -two  years 
in  use.    Very  heav- 
ily incrustated. 

0.05 

1736 

1.00 

0.25 

1.08 
4.29 
10.91 
23.86 

0.80 
1.50 
2.40 
3.50 

50 
44 
45 
46 

1  .041 
0.179 
0.937 
0.958 

New   asphalt-coated 
cast-iron       pipe. 
Rochester,  N.Y.— 
E.Kuichling,1895. 

0.83 

3 

0.75 

1.38 
1  .50 
1  .50 

4.204 
4.234 
4.234 

129.4 
125  .25 
125  .25 

FA 

1.0 

Do. 
1897. 

0.45 

2.27 

4.82 
4.85 

4.128 
4.045 
4.022 

91  .22 
66.84 
66.24 

Do. 

1898. 

0.13 

... 

... 

4.34 
3.76 

4.034 
4.026 

70.23 
75.32 

1  .0 

Do. 

1899. 

0.28 

... 

... 

3.44 
3.25 

4.084 
4.079 

79.79 
81  .93 

1.0 

88 


THE   FLOW   OF   WATER 


TABLE  X.  —  Continued. 


Description  of 
Conduit. 

m 
0.95 

L 

d 

i 
R 

1000s 

V 

c 

a 

7?* 

Old  cast-iron  pipe 
16  years  old,  tuber- 
cles removed.  — 
Fitzgerald. 

... 

4 

1.0 

0  .4167 
1  .241 
1  .8283 

3.723 
4.973 
6.141 

139.1 
141  .1 
143.6 

Cast  iron  intake 
pipe  at  Erie,  Pa., 
8  years  in  use. 

0.48 

8215 

5 

1.25 

... 

0.178 
1.088 

99.8 
102.1 

1.0 

Old  cast-iron  pipe  in 
good  condition, 
some  easy  bends. 
—  Jas.  M.  Gale. 

0.70 

19600 

4 

1 

0.947 

3.458 

112.4 

1  .0 

VII.   GALVANIZED  PIPES,  GLASS,  TIN  AND  LEAD  PIPES. 


New  wrought-iron 
galvanized  pipe, 
straight.  —  Eh- 
mann. 

0.92 

301  .8 

0  .0842 

0.021 

7.61 
29.35 
113  .04 
225.0 
239  .13 

1  .11 
2.13 
3.71 
5.80 
5.90 

87.1 
85.9 
79.2 
84.5 
83.2 

1.016 
1.00 
1  .035 
0.984 
0.970 

Glass  pipe  with  fun- 
nel-mouthpiece. — 
Hamilton  Smith. 

0.87 

63.9 

0  .0764 

0  .0191 

25.01 
50.77 
75.30 
102.6 
129.18 

1  .955 
2.945 
3.685 
4.383 
5.009 

89.5 
94.6 
92.2 
99.3 
100.8 

1.078 
1.140 
1.171 
1.199 
1.219 

Do.,  no  funnel. 

0.84 

63.9 

0  .0764 

0  .0191 

17.97 
132  .51 

1  .398 
4.373 

83.6 
96.3 

1  .030 
1  .185 

Glass  pipe,  straight. 
—  Darcy. 

0.84 

147.0 

0.163 

0  .0407 

0.96 
7.71 
57.62 
111  .91 

0.502 
1  .591 
4.849 
6.916 

80.3 
89.8 
100.1 
102.4 

0.930 
1.044 
1.164 
1.191 

New  lead  pipe, 
straight.  —  Darcy. 

0.84 

172 

0  .0886 

0  .0221 

0.44 
8.14 
54.36 
146  .32 

0.213 
1  .089 
3.35 
5.509 

68.3 
81  .1 
96.5 
96.8 

0.843 
1  .00 
1  .191 
1  .195 

New      lead      pipe, 
straight.  —  Darcy. 

0.84 

172 

0  .1345 

0  .0336 

0.82 
7.48 
56.00 
158  .82 

0.394 
1.404 
4.318 
7.562 

75.0 
86.8 
99.5 
103.5 

0.90 
1.038 
1  .178 
1  .238 

Tin  pipe,  straight. 
—  Dubuat. 

0.98 

0  .0888 

0  .0222 

0.196 
0.641 
3.91 
5.39 
7.54 
9.91 
13.7 
29.82 
30.31 
99.01 

0.141 
0.322 
0.772 
0.927 
1  .183 
1.342 
1  .476 
2.546 
2.606 
5  .223 

67.6 
85.3 
82.8 
84.8 
91  .4 
90.5 
92.9 
98.9 
100.4 
111  .4 

0.746 
0.942 
0.914 
6.937 
1.009 
1.00 
1  .015 
1  .092 
1  .109 
1  .120 

DESCRIPTION   OF   CONDUIT,   ETC. 


89 


TABLE  X. — Continued. 


Description  of 
Conduit. 

m 

L 

d 

R 

1000  s 

V 

t 

a 

Tin  pipe.     Straight. 

0.95 

192 

0  .1184 

1  .0296 

5.40 

1.116 

88.2 

1  .00 

—  Bossut. 

192 

u 

" 

10.76 

1.678 

94.0 

1  .068 

64 

({ 

" 

15.08 

2.075 

98.2 

1  .116 

32 

(C 

tt 

26.94 

2.946 

104.3 

1  .185 

32 

ft 

(t 

52.98 

4.31 

108.8 

1.234 

Do. 

0.94 

63 

0.1184 

0  .0296 

113.4 

6.143 

106.0 

1  .220 

126 

« 

n 

113.5 

6.15 

106.1 

1  .220 

189 

tt 

u 

113.4 

6.157 

106.2 

1  .220 

VIII.   PIPES  AND  OPEN  CONDUITS  OF  PLANED  OR  ROUGH  BOARDS. 


Redwood  stave  pipe. 
Los  Angles,  Cal.  — 
A.  L.Adams,1898. 

0.93 

4-8000 

1.166 

0.292 

0.17 
0.161 
0.178 
0.145 
0.391 
0.638 
1.355 

0.698 
0.698 
0.751 
0.691 
1.167 
1.531 
1.181 

99 
101 
104 
105 
109 
112 
113 

0.908 
0.926 
0.953 
0.963 
1.01 
1.027 
1.043 

... 

... 

Do. 
At    Astoria,    Ore,  — 
A.  L.  Adams. 

0.96 

4188 

1  .5 

0.375 

2.07 

3.605 

132.9 

jri 

Wooden  stave  pipe, 
at  Cedar  River, 
Wash.  Long  but 
easy  curves.  Sev- 
eral years  in  use. 
Some  slight  de- 
posits. —  Theron  A. 
Noble. 

0.50 

3.67 

0.917 

1.067 
1.134 
1  .191 
1  .262 
1  .33 
1  .331 
1  .401 
1  .627 
1  .757 
1.757 
1.888 

3.468 
3.522 
3.685 
3.853 
3.964 
3.972 
4.072 
4.415 
4.595 
4.635 
4.831 

110.1 
108.6 
110.9 
112.6 
112.9 
113.1 
112.9 
113.7 
113.8 
114.8 
115.6 

V* 

... 

... 

Do. 

0.58 

4.5 

1.125 

0.342 
0.342 
0.436 
0.558 
0.557 
0.672 
0.783 
0.856 
0.983 
1.076 
1  .162 

2.282 
2.276 
2.65 
3.07 
3.05 
3.41 
3.724 
3.924 
4.215 
4.42 
4.69 

116.8 
115.8 
119.4 
122.1 
121  .4 
123.7 
125  .2 
126.2 
126.5 
126  .7 
129.2 

V* 

90 


THE   FLOW   OF  WATER 


TABLE  X.  —  Continued. 


Description  of 
Conduit. 

m 
0.51 

L 

d 

R 

1000  s 

V 

c 

a 

Wooden  stave  pipe 
at  Ogden,  Utah. 
Many  easy  curves 
two  years  in  use 
—  Marx-Wing  anc 
Hoskins. 

4.000 

6.0 

1.5 

1  .40 
1.68 
2.14 
2.43 
2.96 
3.59 
3.63 

110. 
112. 
115. 
119. 
122. 
126. 
124. 

y\ 

Rectangular  pipes 
of  unplaned  board. 
—  Darcy. 

0.68 

145.7 

0.319 

0.523 
1.067 
1.933 
2.733 

3.867 
6.267 
7.267 
8.80 

1.23 

1.778 
2.267 
2.939 
3.529 
4.349 
4.625 
5.307 

94. 
96. 
96. 
99. 
100. 
97. 
96. 
100. 

yA 

... 

... 

... 

Rectanglar  pipe  of 
unplaned  boards. 
—  Darcy. 

0.73 

230 

0.505 

0.475 
1.076 
1.90 
2.91 
4.27 
5.06 
5.76 
6.61 

1.67 
2.52 
3.37 
4.23 
5.07 
5.52 
5.91 
6.37 

107. 
108. 
108. 
110. 
109. 
109. 
109.7 
110.3 

FrV 

Provo  Canal  Flume, 
Utah.  Semicircu- 
lar conduit  of 
planed  staves, 
several  years  in 
use.—  W.  B.  Mc- 
Dougall.  w^,  P<fp 

0.85 
0.81 

... 

... 

1.45 
1  .46 

1.0 
1.0 

5.67 
5.37 

147.7 
141.8 

v* 

yvt 

Wooden  trough, 
trapezoidal.  Bot- 
tom width  10.4 
ft.  Rittinger. 

0.71 

W 

0.24 
0.26 
0.38 
0.41 

0.159 
0.173 
0.237 
0.246 

34.3 
it 

a 
it 

8.26 
8.21 
10.11 
10.64 

111  .9 
106.6 
112.1 
115.8 

vrk 

Rectangular         test 
channel  of  planed 
boards.  —  Darcy- 
Bazin.    Series  28. 

0.94 

0.328 

0.04 
0.08 
0.11 
0.14 
0.17 
0.20 
0.22 

0.029 
0.052 
0.066 
0.075 
0.084 
0.091 
0.093 

4.7 
d 

tt 

ti 
ii 
it 
tt 

0.90 
1.30 
1  .58 
1  .74 
1  .94 
2.11 
2.16 

76.5 
83.0 
89.4 
92.7 
97.6 
102.1 
103.2 

rh 

DESCRIPTION   OF  CONDUIT,  ETC. 


91 


TABLE  X.  —  Continued. 


Description  of 
Conduit. 

m 

W 

d 

R 

1000s 

V 

c 

a 

Semicircular        test 
channel     of     un- 
planed        boards. 
—   Darcy-Bazin. 
Series  26. 

0.70 

3.16 
3.62 
3.89 
4.08 
4.24 
4.33 
4.43 
4.48 
4.53 
4.56 
4.59 
4.59 
4.59 

0.63 
0.88 
1.07 
1.24 
1  .40 
1  .55 
1.68 
1  .79 
1.92 
2.02 
2.14 
2.24 
2.29 

0.39 
0.537 
0.632 
0.717 
0.796 
0.856 
0.926 
0.964 
1  .005 
1.054 
1  .096 
1.129 
1.148 

1.5 
tt 

ii 
n 
n 
n 

1C 

ii 
a 
if 
it 
a 
a 

2.61 
3.23 
3.71 
4.04 
4.25 
4.51 
4.64 
4.87 
5.0 
5.18 
5.29 
5.45 
5.54 

107.8 
113.8 
120.6 
123.0 
123.2 
125.8 
124.7 
128.2 
128.2 
130.3 
130.4 
132.3 
133.5 

yA 

Rectangular         test 
channel     of     un- 

0.66 

6.53 

0.26 
0.41 

0.24 
0  .363 

2.08 

u 

2.08 
2.69 

93.2 

97.8 

yh 

planed   boards.  — 

0.53 

0  .453 

(( 

3  .16 

102.8 

Darcy-Bazin. 

0.63 

0.528 

(t 

3  .53 

106.5 

Series  6. 

0.73 

0.601 

(I 

3  .78 

106.9 

0.81 

0.648 

(t 

4  .13 

112.5 

... 

0.90 
0  .99 

0.704 
0.759 

11 
11 

4.34 
4.51 

113.5 
113.5 

... 

1  .06 
1  .14 

0.801 
0.846 

It 
(t 

4.72 

4.88 

115.8 
116.3 

... 

1  .20 

0.880 

tl 

5.09 

119.0 

1.28 

0.992 

tl 

5.21 

118.9 

... 

Do. 

Series  7. 

0.70 

6.53 

0.20 
0.30 
0.38 

0.188 
0.272 
0.342 

4.9 
tt 

a 

2.71 
3.70 
4.35 

89.3 
101.2 
106.2 

FA 

... 

0.46 
0.53 
0.60 
0.66 
0.72 
0.78 
0.83 
0.89 
0.94 

0.402 
0.453 
0.504 
0.547 
0.587 
0.628 
0.662 
0.698 
0.727 

a 
tt 
tt 
tt 
tt 
tt 
tt 
it 
ti 

4.85 
5.29 
5.61 
5.93 
6.23 
6.45 
6.71 
6.90 
7.15 

109.4 
112.2 
113.0 
114.5 
116.1 
116.4 
117.8 
117.9 
119.8 

... 

Rectangular        test 
channel     of     un- 
planed        boards. 
Darcy-Bazin. 
Series  8, 

0.71 

6.53 

0.15 
0.25 
0.32 
0.38 
0.45 
0.50 
0.54 
0.60 
0.65 
0.69 
0.74 
0.78 

0.147 
0.231 
0.289 
0.341 
0.393 
0.431 
0.466 
0.506 
0.541 
0.572 
0.604 
0.630 

8.24 
it 

tt 
tt 
tt 
tt 
tt 
tt 
tt 
tt 
tt 
tt 

3.52 

4.42 
5.23 
5.83 
6.24 
6.74 
7.07 
7.44 
7.73 
8.03 
8.26 
8.57 

100.4 
101  .4 
107.1 
109.8 
109.7 
113.1 
115.8 
115.2 
115.8 
116.9 
117.1 
119.0 

yiV 

92 


THE   FLOW   OF   WATER 


TABLE  X.  —  Continued. 


Description  of 
Conduit. 

m 

w 

d 

R 

1000s 

V 

c 

a 

Triangular           test 

0.69 

1.85 

0.92 

0.327 

4.9 

4.13 

103.1 

TA 

channel      of     un- 

2.39 

1.19 

0.422 

' 

5.02 

110.4 

. 

planed  boards.  — 

2.79 

1.40 

0.494 

t 

5.56 

113.0 

Darcy-Bazin. 

•     .     4 

310 

1.55 

0.549 

c 

6.03 

116.2 

Series  23. 

.     .     . 

3.38 

1.69 

0  .597 

1 

6.36 

117.6 

.     .     . 

3.64 

1.82 

0.643 

( 

6.59 

117.3 

.     .     . 

3.86 

1  .93 

0.683 

f 

6.83 

118.0 

.     .     . 

4.07 

2.03 

0.719 

(t 

7.03 

118.4 

.     .     . 

4.26 

2.13 

0.752 

tt 

7.23 

119.0 

.     .     . 

4.43 

2.22 

0.783 

It 

7.40 

119.5 

.     .     . 

4.61 

2.30 

0.814 

(I 

7.54 

119.4 

4.75 

2.37 

0.839 

t( 

7.75 

120.9 

Flume  of  Kern  River 

0.72 

8.0 

2.5 

1.538 

1.5 

6.681 

139.1 

r* 

Power   Plant    No. 

3.0 

1.714 

tt 

6.968 

137.4 

. 

1.     Plain    boards, 

3  5 

1  866 

d 

7.394 

139.8 

seams     c  o  v  e  r  e  c 

4.0 

2.00 

ti 

7.920 

144.8 

. 

with   ^    inch    bat- 

tens.     Sect,    rect 

Length  1029.  6  feet 

Tunnel  lined  with 

cement        plaster 

1     cement     to    2 

sand  of  same  sec- 

tion and  slope  be 

low  flume. 

—  F.  C.  Finkle. 

IX.   PIPES  AND  OPEN  CONDUITS  OF  CEMENT  OR  CONCRETE. 


Cement  lined  pipe  of 
of   wrought    iron 

0.83 

L 

8171 

1  .667 

0.416 

0.23 
0.44 

0.949 
1  .488 

97.4 
109  8 

0.92 
1  .046 

Three-stop  valves 
and    two    large 

... 

0.73 
1  .04 

1.925 
2.329 

110.7 
112.0 

1  .054 
1  .066 

branches  on  line 

1  .34 

2.598 

110.1 

.046 

—  Fanning 

1  58 

2  867 

111  7 

063 

1  99 

3  271 

113  5 

081 

. 

.  .  . 

.  .  . 

2.28 
2.72 
3.0 
3  20 

3.439 
3.741 
3.920 
4  040 

111  .7 
111  .1 
110.8 
110  .6 

.063 
.056 
1.052 
1  .051 

Test    pipe    of    clear 
cement       Diame- 

0.95 

2.624 

0.656 

0.625 
1  .05 

2.78 
3.65 

137.1 
139.2 

•p?* 
1  .139 

ter  0  8   meter.  — 

1  .375 

4.20 

139  .5 

1  .140 

Dijon. 

1  .725 

4.72 

140.4 

1  .141 

Quoted  by  Bazin. 

1  .75 

4.79 

141  .2 

1  .155 

... 

1.88 
2  57 

4.92 
5  81 

141  .4 
141  .4 

1  .157 
1  .157 

3.27 

6.58 

142.5 

1.166 

DESCRIPTION   OF   CONDUIT,   ETC. 


93 


TABLE  X.  —  Continued. 


Description  of 
Conduit. 

m 

W 

d 

R 

1000s 

V 

c 

a 

Semicircular       test- 

1.0 

2.874 

0.59 

0.366 

1  .5 

3.02 

128.9 

ytV 

channel    of    clear 

. 

3.294 

0.83 

0.503 

3.82 

135.6 

cement.  —  Darcy- 

3.563 

1  .03 

0.605 

4.16 

138.0 

Bazin.    Series  24. 

.  .  . 

3.707 

1.18 

0.682 

4.60 

143.7 

3.832 

1.34 

0.750 

4.87 

145.1 

3.924 

1  .47 

0.809 

5.12 

147.1 

3.97 

1  .61 

0.867 

5.29 

146.7 

4.05 

1  .72 

0.915 

5.51 

148.8 

4.075 

1.83 

0.949 

5.75 

152.5 

4.095 

1  .94 

0.992 

5.91 

153.3 

4.101 

2.05 

1.029 

6.06 

154.2 

4.16 

2.08 

1.034 

6.11 

155.1 

Rectangular        test 

0.95 

5.94 

0.18 

0.168 

4  9 

3.34 

116.5 

rh 

channel    of    clear 

. 

0.28 

0.251 

4.39 

125.1 

cement.  —  Darcy- 

0.36 

0.322 

5.04 

126.9 

Bazin.     Series  2. 

0.43 

0.375 

5.68 

132.4 

0.56 

0.43 

6.06 

132.4 

0.56 

0.475 

6.51 

135  .1 

0.63 

0.518 

6.83 

135.5 

0.69 

0.558 

7.12 

136.2 

0.76 

0.595 

7.41 

137.2 

0.80 

0.632 

7.63 

137.2 

0.86 

0.665 

7.86 

137  .8 

... 

... 

0.91 

0.696 

8.07 

138.2 

Sudbury    conduit. 

0.94 

8.6 

3.071 

1  .863 

0  .1606 

2.529 

146.2 

v* 

Plaster    of    pure 

3.574 

2.048 

0  .1596 

2.672 

147.9 

cement  over  brick 

3.768 

2.111 

0  .1580 

2.805 

153.9 

work.        Sides 

nearly       vertical, 

bottom    flat    seg- 

mental     arch.  — 

Fteley  &  Stearns. 

Aqueduct  of  the  Se- 

0.98 

5.38 

1  .41 

0.50 

4.06 

152  .5 

FrV 

rino,  Naples.  Pure 

cement,    polished. 
Sides  vertical,  bot- 

tom        ellyptical 

arch.    —  Perrone, 

1896. 

94 


THE   FLOW  OF   WATER 


TABLE  X.  —  Continued. 


Description  of 
Conduit. 

m 

W 

d 

R 

1000s 

V 

c 

a 

Semicircular        test 

0.85 

2.913 

0.61 

0.379 

1  .5 

2.89 

120.5 

FA 

channel  of  cement 

3.36 

0.88 

0.529 

3.43 

122.0 

mortar.     Two- 

3.616 

.09 

0.635 

3.89 

125.2 

thirds     cement, 

3.760 

.24 

0.706 

4.30 

132.1 

one-third  fine  sand. 

3.891 

.41 

0.787 

4.51 

131.3 

—  Darcy-Bazin. 

. 

3.963 

.54 

0.839 

4.80 

135.3 

. 

Series  25. 

. 

4.029 

.69 

0.900 

4.94 

134.5 

. 

•4  .068 

.80 

0.941 

5.20 

138.3 

4.088 

1  .92 

0.983 

5.38 

140.1 

4.095 

1  .98 

1.006 

5.48 

141  .0 

4.095 

2.04 

1.022 

5.55 

141.7 

4.095 

2.09 

1.038 

5.56 

143.5 

Conduit     of     North 

0.83 

1.02 

0.619 

0  .333 

1  .58 

110 

yrz 

Metropolitan  Sew- 

. !  . 

1  .52 

0.928 

" 

2.21 

126 

age  System,  East 

2.04 

1.208 

tt 

2.70 

134 

Boston        section 

2.45 

1.408 

" 

3.03 

139 

Brickwork  washec 

3.16 

1.830 

« 

3.48 

141 

with  cement.  Sec- 

. !  . 

!  !  ! 

3.75 

1.999 

" 

3.73 

145 

tion    circular,    di- 

4.62 

2.31 

tt 

4.18 

150 

ameter  9  ft.—  Th 

Horton.       Experi- 

ments of  1896.     10 

months  in  use. 

Do. 

Experiments  of  1897. 

0.59 

2.15 

1.28 

0.333 

2.55 

123 

YTZ 

2.74 

1.56 

" 

2.90 

127 

3.19 

1.76 

" 

3.06 

126 

3.20 

1  .97 

" 

3.18 

131 

Do. 

0.56 

1  .99 

1  .12 

0.333 

2.38 

119 

yh 

Experiments  of  1900. 

2.83 

1  .61 

2.82 

121 

. 

3.64 

1  .95 

" 

3.16 

124 

4.18 

2.13 

" 

3.30 

124 

... 

Do. 

0.65 

6.0 

1.02 

0.688 

0.50 

1.99 

107 

rli 

Charlestown  section. 

1  .44 

0.958 

tt 

2.46 

112 

Sides    vertical,    bot- 

1.91 

1.187 

tt 

2.825 

115 

tom  flat  arch. 

2.40 

1.387 

" 

3.33 

118 

Experiments  of  1896. 

.  .  '. 

2.89 

1.539 

" 

3.44 

124 

'.  '.  '. 

10  months  in  use. 

Do. 

0.40 

2.91 

1  .54 

0.5 

2.97 

107 

rA 

Experiments  of  1897. 

... 

3.29 

1  .64 

3.16 

111 

... 

s\i! 


DESCRIPTION   OF   CONDUIT,    ETC. 


95 


TABLE  X. — Continued. 


Description  of 
Conduit. 

m 

W 

d 

R 

1000s 

V 

c 

a 

Conduit     of     North 

0.33 

2.29 

1.34 

0.50 

2.66 

102 

v* 

Metropolitan  Sew- 

2.78 

1  .51 

u 

2.86 

104 

age       System, 

3.20 

1.64 

It 

3.04 

106 

Charlestown     sec- 

tion (continued). 

Experiments  of  1900. 

Aqueduct    of    Glas- 

0.80 

9.6 

.22 

0.182 

1  .67 

125.0 

rA 

gow.       Smooth 

.47 

2.07 

126.3 

.  .  . 

concrete.  Nearly 

.47 

2.10 

128.5 

rectangular,     bot- 

.49 

2.21 

134.5 

tom   flat    arch.  — 

.50 

2.13 

129.3 

Fairlie      Bruce. 

1  .50 

2.15 

130.3 

1896. 

. 

. 

1.55 

2.17 

129.4 

. 

. 

1.60 

2.20 

129.3 

1.61 

2.23 

130.5 

1.61 

2.22 

129.7 

1.62 

2.24 

130.6 

1.63 

2.25 

130.8 

1.74 

2.26 

126.9 

1.81 

2.41 

135.4 

... 

Millrace     at     Idria, 

0.65 

2.04 

0.977 

0.5 

2.523 

114.1 

v& 

Hungary.  Cement 

mortar  on  rubble 

masonry.   —  Rit- 

tinger. 

Aqueduct  of  Roque 

0.72 

6.88 

1  .504 

3.72 

10.26 

137 

v* 

favour,    canal    of 

Marseilles.  Bottom 

of    clear    cement, 

sides  of  good  brick 

work.     Rectangu- 

f 

lar.  —  Baumgarten. 

Aqueduct      of     the 

0.58 

66.0 

5.12 

0.11 

3.52 

148.5 

vb 

Cervo,  Canal    Ca- 

. 

5.76 

« 

3.76 

149.3 

vour.     Bottom  of 

. 

7.20 

d 

4.38 

155.8 

good      concrete, 

sides    of   brick. 

Rectangular.  — 

Passini  &  Gioppi, 

T892. 

96 


THE   FLOW   OF  WATER 

TABLE  X.  —  Continued. 


Description  of 
Conduit. 

m 

W 

d 

R 

1000s 

V 

c 

a 

Gage  Canal.        San 
Bernardino,     Cal. 
Channels  in  earth, 
roughly  coated 
with  cement  plas- 
ter, 1  part  cement, 
4  parts  sand.  — 
U.S.Geol.  Survey. 

0.49 
0.48 
0.46 
0.48 
0.44 
0.47 

9.25 
10.25 
14.0 
12.25 
16.0 
17.0 

3.5 
3.5 
3.5 
5.5 
3.5 
3.5 

1.82 
1.94 
2.13 
2.13 
2.38 
2.38 

0.4 
0.4 
0.478 
0.382 
0.520 
0.413 

3.14 

3.28 
3.78 
3.38 
4.24 

3.78 

117 
117 
119 
119 
120 
121 

rA 

Canal   of  Verona. 
Channel     lined 
with    Beton    ma- 
sonry.         Trape- 
zoidal.       Bottom 
width  about  20  ft. 

0.00 

5.12 

0.31 

4.2 

107.8 

yA 

San  Gabriel  Tunnel 
No.  15. 
Coated  with  cement 
mortar,  1  cement 
to  3  sand.     Sec- 
tion rect.    L.  446 
ft.  —  Lippincott. 

0.89 

4.5 

4.0 

1.31 

0.96 

5.01 

141.3 

yA 

San  Gabriel  Tunnel 
No.  23. 
L.  318  ft. 

0.89 

... 

... 

1.37 

0.86 

4.74 

141.6 

yA 

Old  Aqueduct  of  Los 
Angeles. 
Coated  with  cement 
plaster     on     con- 
crete.    4  years  in 
use.     Covered. 

0.95 
0.90 

... 

0.817 
0.830 

0.51 
0.51 

2.71 
2.81 

132.6 
136.7 

yA 

Colton  Canal. 
Channel  lined  with 
concrete.    Bottom 
clean,  sides  lightly 
coated  with  moss. 

0.26 

0.98 

20.70 

2.27 

86.7 

yA 

Santa  Ana  Canal. 
Channel  lined  with 
concrete.    Bottom 
covered  with  sand 
and    gravel,    sides 
coated  with  plants 
in  places. 

0.33 

0.817 

1.06 

2.62 

89.2 

yA 

DESCRIPTION   OF   CONDUIT,   ETC. 


97 


TABLE  X. — Continued. 


Description  of 
Conduit. 

m 

W 

d 

R 

1000s 

V 

c 

a 

Riverside    canal. 
Open    conduit 
coated  with  cement 
mortar  on  concrete. 
Bottom      covered 
with  fine  sand  to 
a  depth  of  1.5  to 
2.5    ft.         Trape- 
zoidal. 

-0.27 
-0.08 

... 

.... 

1.49 
0.703 

0.92 
0.63 

1.96 
1.22 

52.9 
38.0 

1 

vh 

X.  BRICK  CONDUITS. 


Sudbury       Conduit. 

0.80 

9.0 

4.672 

2.359 

0  .03341 

1.207 

136.0 

v& 

Hard  glazed  brick, 

4.972 

2.417 

0  .0488 

1  .497 

137.9 

smoothly  jointed, 

3.319 

1  .963 

0  .0625 

1  .512 

136.5 

fairly  clean.  Sides 

2.561 

1  .648 

0  .0948 

1  .616 

129.3 

nearly       vertical, 

2.998 

1  .838 

0.1155 

1  .983 

136.1 

bottom  flat  arch. 

3.369 

1  .981 

0  .1356 

2.255 

137.6 

Fteley  &  Stearns. 

2.192 

1  .468 

0  .1466 

1  .931 

131  .6 

4.602 

2.343 

0  .1793 

2.889 

141  .0 

3.878 

2.151 

0  .2102 

2.955 

139.0 

3.266 

1.943 

0  .2389 

2.957 

137.3 

. 

.  . 

1  .799 

1.251 

0  .2553 

2.448 

137.0 

'.  '.  '. 

2.245 

1.495 

0  .2580 

2.687 

138.5 

_ 

2.707 

1.714 

0  .2602 

2.886 

136.6 

2.881 

1.789 

0  .4604 

4.163 

142.9 

... 

... 

3.437 

2.005 

0  .4913 

4.913 

140.8 

... 

New    Croton   Aque- 

0.68 

13.6 

0.75 

0  .1326 

1  .1 

110.4 

r* 

duct,   New   York. 

1  .0 

a 

1.37 

118.9 

Good     brickwork. 

1.25 

it 

1.59 

123.0 

Sides  nearly  verti- 

1 .50 

t( 

1.80 

127.4 

cal,    bottom    flat 

1.75 

(t 

1.94 

128.3 

arch.  —  Fteley, 

2.0 

it 

2.1 

129.2 

1895. 

. 

. 

2.25 

(t 

2.27 

131.2 

. 

. 

2.50 

it 

2.40 

132.1 

. 

.  .  . 

2.75 

tt 

2.52 

132.1 

.  .  . 

3.0 

is 

2.65 

133.0 

. 

.  . 

3.50 

ii 

2.89 

134.0 

... 

... 

3.81 

d 

3.02 

134.0 

Rectangular        test 

0.57 

6.27 

0.20 

0.192 

4.9 

2.75 

89.7 

yrs 

channel    of    com- 

0.31 

0.284 

n 

3.66 

98.3 

mon      brickwork 

0.41 

0.365 

tt 

4  .18 

98  8 

rather     rough.  — 

0.49 

0.424 

it 

4.72 

103.7 

Darcy-Bazin 

. 

0.57 

0.481 

d 

5.10 

105.1 

Series  3. 

.  . 

0.65 

0.540 

it 

5.33 

103.7 

.  .  . 

.  .  . 

0.71 

0.580 

IS 

5.68 

106.3 

0.77 

0.620 

11 

6.01 

109.0 

.  .  . 

.  .  . 

0.85 

0.668 

ss 

6.15 

107.4 

.  .  . 

.  .  . 

0.90 

0.697 

tt 

6.47 

110.8 

0.97 

0.739 

SI 

6.60 

109.7 

1.04 

0.779 

11 

6.72 

108.7 

98 


THE  FLOW  OF  WATER 


TABLE  X.  —  Continued. 


Description  of 
Conduit. 

m 

L 

d 

R 

1000s 

V 

c 

128.3 
125.1 
124.5 
124.9 
127.9 
122.9 
122.7 
124.3 
124.2 

a 
1.0 

Brick  sewer  at  Mil- 
waukee, Wis. 
Smooth  brick,  well 
pointed.  —  G.  H. 
Benzenberg. 

0.57 

2534 

12.0 

3.0 

0.523 
0.547 
0.814 
0.821 
0.793 
1  .046 
1.046 
1.040 
1.010 

5.083 
5.043 
5.154 
6.195 
6.207 
6.886 
6.872 
6.961 
6.821 

Brick  sewer,  Dor- 
chester Bay  Tun- 
nel. Inverted  sy- 
phon. Hard  brick, 
well  pointed. 
Sewer  slime.  — 
Clarke,  1895. 

0.47 

7166 

7.5 

1.875 

0.513 
0.554 
0.581 

3.769 
3.798 
3.929 

121.0 
118.0 
119.0 

1.0 

Syphon  Aqueduct  of 
the    Elvo,    Canal 
Cavour.    This  con- 
duit consists  of  5 
oval    tubes,    each 
having  a  cross  sec- 
tion of  119.25  sq. 
ft.  —  Passini   & 
Gioppi,  1892. 

0.40 

581.5 

2.78 

0.067 
0.107 
0.152 
0.208 
0.276 
0.361 
0.462 
0.586 
.733 

1  .61 
1  .95 
2.31 
2.66 
3.1 
3  .53 
4.01 
4.52 
5.094 

117.3 
112.9 
112.2 
111  .6 
111.8 
111.6 
111.8 
112.0 
113.1 

f.6 

XI.  CHANNELS  LINED  WITH  ASHLAR  OR  RUBBLE  MASONRY. 


Chazilly  Canal.  Ash- 
lar           masonry, 
smoothly  dressed, 
section     trapezoi- 
dal, very  regular. 
—  Darcy-Bazin. 
Series  39. 

0.57 

W 
4.04 
4.10 
4.14 
4.18 

0.50 
0.78 
1.0 
1.20 

0.41 
0.57 
0.68 
0.77 

8.1 

u 
(I 
(( 

5.73 
7.52 
8.19 

8.75 

100.0 
111  .0 
110.0 
111  .0 

TnV 

Aqueduct    of    Crau, 
Canal  of  Craponne. 
Ashlar     masonry, 
smoothly  dressed. 
Section   rectangu- 
lar. —  Darcy-Ba- 
zin. 

0.59 

8.5 

3.0 

1.774 

0.84 

5.55 

125.0 

V& 

DESCRIPTION  OF  CONDUIT,   ETC. 


99 


TABLE  X.  —  Continued. 


Description  of 
Conduit. 

ra 

W 

d 

R 

1000s 

V 

c 

a 

Solani        Aqueduct. 

0.44 

85.0 

2.66 

2.52 

0.151 

2.20 

112.8 

1.0 

Ganges  Canal,  In- 

2.88 

2.72 

0.145 

2.54 

117.9 

1  .04 

dia.     Rectangular 

3.13 

2.94 

0.20 

2.51 

103.5 

0.90 

conduit  consisting 

3.12 

2.94 

0.208 

2.79 

112.8 

0.984 

of    two    sections, 

. 

.  .  . 

3.18 

2.99 

0.253 

3.20 

116.4 

1.016 

separated     by     a 

.  .  . 

.  .  . 

3.96 

3.65 

0.473 

4.83 

116.2 

1.0 

central   wall,  len- 

. .  . 

.  .  . 

4.60 

4.20 

0.025 

1.24 

121.0 

0.99 

gth  920  ft.     Floor 

of  brick,  laid  flat, 

sides  of  masonry. 

Some         deposits 

here     and     there. 

Right    section.  — 

Allan       Cunning- 

ham, 1880. 

New  and  well  built 

0.32 

6.0 

0.50 

0.32 

42.35 

9.45 

80.5 

FA 

channel     of     dry 

0.32 

46.42 

10.49 

85.4 

rubble  masonry  of 

0.57 

0.37 

42.35 

9.89 

82.5 

. 

large  stones.  Semi- 

6.37 

46.42 

10.97 

83.6 

. 

circular.  —  Kutter, 

1867. 

Old  channel  of  dry 

0.27 

8.0 

0.55 

0.36 

82.8 

11.81 

68.6 

1.0 

rubble  masonry  oi 

8.0 

0.55 

0.38 

99.3 

13.32 

68.4 

« 

large   stones,   bed 

. 

7.4 

0.60 

0.39 

106.8 

13.75 

67.3 

tt 

somewhat      dam- 

10.6 

0.90 

0.58 

82.8 

15.54 

70.6 

d 

aged.     Semicircu- 

10.6 

0.90 

0.61 

99.3 

18.28 

72.8 

u 

lar.  —  Kutter, 

9.0 

1.0 

0.65 

106.8 

19.17 

72.8 

0.97 

1867. 

Spillway  of  Grosbois 

0.15 

5.98 

0.36 

0.324 

101  .0 

12.29 

67.9 

FrV 

Reservoir.       Ash- 

6.01 

0.55 

0.467 

n 

16.18 

74.5 

.   .   . 

lar    with    cement 

6.05 

0.71 

0.580 

ii 

18.68 

77.2 

. 

joints,  partly  dam- 

6.07 

0.84 

0.662 

tt 

21.09 

81.6 

.   .   . 

aged,  covered  with 

a     sticky     slime. 

Rectangular.  — 

D  a  rcy-Bazin. 

Series  32. 

Do. 

0.16 

6.0 

0.49 

0.424 

37.0 

9.04 

72.2 

FrV 

Tailrace     of     Gros- 

6.1 

0.77 

0.620 

(f 

11  .46 

75.7 

bois        Reservoir 

6.1 

0.97 

0.745 

tt 

13.55 

81  .6 

—  Darcy-B  a  z  i  n 

6.1 

1.16 

0.846 

ti 

15.08 

84.9 

Series  33. 

Tailrace  of  dry  rub- 

-0.03 

0.42 

0.289 

2.5 

1  .257 

46.8 

1.012 

ble  masonry,  pav- 

0.56 

0.359 

tt 

1  .491 

49.8 

1.015 

ed,  semicircular.  — 

0.69 

0.419 

tt 

1.643 

50.8 

1.010 

Rittinger,      1855 
Quoted  by  Kutter 

100 


THE   FLOW   OF   WATER 


TABLE  X.  —  Continued. 


Description  of 
Conduit. 

m 

w 

d 

R 

1000  s 

V 

c 

a 

Aqueduct  of  dry 
rubble  masonry, 
paved.  Rectan- 
gular. Rittinger. 

-0.03 

... 

0.24 
0.65 
0.81 

0.213 
0.439 
0.486 

4.5 

a 

n 

1  .324 
2.396 
2.432 

42.8 
53.9 
52.0 

1.012 
1.056 
1.0 

Do. 

Tailrace  of  dry  rub- 
ble masonry  paved. 
Trapezoidal.  Ritt. 

-0.02 

0.35 
0.47 
0.56 

0.278 
0.351 
0.403 

3.6 
tt 

n 

1.502 
1.928 
2.104 

47.5 
54.2 
55.2 

1.022 
1  .08 
1  .041 

Headrace  of  dry 
rubble  masonry 
paved.  Trapezoid- 
al. —  Rittinger. 

0.04 

... 

0.26 
0.45 

0.213 
0.344 

3.8 
« 

1  .369 
1  .828 

48.1 
50.6 

1.012 
0.988 

Grosbois  Canal. 
Channel  of  rough- 
ly hammered 
stone  masonry.  — 
Darcy-Bazin. 
Series  1. 

0.15 

3.9 
3.6 
3.5 
3.5 

1  .6 
1  .5 
1.2 
0.9 

0.88 
0.84 
0.71 
0.62 

12.1 
14.0 
29.0 
60.0 

9.58 
8.36 
11.23 
13.93 

73.5 
77.3 

78.4 
72.5 

1  .0 
1  .06 
1.12 
1.055 

Do. 
Masonry  in  bad  con- 
dition,   mud    and 
stones   in   bed.  — 
Darcy-Bazin. 
Series  46. 

-0.02 

6.8 
6.9 
6.9 
7.0 

1.5 
2.0 
2.4 
2.4 

0.88 
1  .23 
1.40 
1  .42 

0.648 
0.671 
0.683 
0.683 

1.47 
2.02 
2.34 

2.78 

62 
70 
76 

87    • 

1.02 
1.06 
1.02 
1  .024 

XII.  CHANNELS  LINED  WITH  PEBBLES  HELD  IN  PLACE  WITH  CEMENT. 


Semicircular       test- 

0.38 

3.1 

0.7 

0.454 

1.5 

2.17 

83.1 

•pTffr 

channel  lined  with 

3.4 

0.9 

0.546 

t 

2.50 

89.4 

pebbles        f       to 

3.5 

1  .1 

0.619 

( 

2.69 

88.2 

1-inch      diameter, 

3.7 

1  .2 

0.681 

t 

2.93 

89.5 

held  in  place  with 

3.8 

1  .3 

0.731 

( 

3.05 

92.1 

cement.  —  Darcy- 

3.8 

1  .4 

0.784 

i 

3  .22 

93.9 

Bazin.     Series  27. 

. 

3.9 

1  .5 

0.826 

t 

3.33 

94.6 

. 

4.0 

1  .7 

0.900 

i 

3.54 

96.3 

. 

4.0 

1.9 

0.968 

tt 

3.73 

97.9 

... 

4.0 

2.0 

1.012 

u 

3.95 

102.1 

... 

Do. 

0.19 

6.0 

0.27 

0.25 

4.9 

2.16 

61  .7 

rh 

Section  rectangular. 

0.41 

0  .357 

< 

2.95 

70.5 

—  Darcy-Bazin. 

0.53 

0.450 

( 

3.40 

72.5 

Series  4. 

0.63 

0  .520 

t 

3.84 

76.1 

0.73 

0.588 

t 

4.14 

77.2 

0.82 

0.644 

t 

4.43 

78.8 

0.91 

0.740 

i 

4.64 

79.3 

0.99 

0.746 

i 

4.88 

80.7 

.  .  . 

.  .  . 

1  .06 

0  .785 

i 

5.12 

82.6 

.  .  . 

.  .  . 

1  .15 

0.832 

( 

5.26 

82.4 

.  .  . 

.  .  . 

1.23 

0.871 

( 

5.43 

83.1 

.  .  . 

1  .30 

0  .910 

' 

5.57 

83.4 

.  .  . 

DESCRIPTION   OF   CONDUIT,    ETC. 


101 


TABLE  X.  —  Continued. 


Description  of 
Conduit. 

m 

W 

d 

R 

1000s 

V 

c 

a 

Rectangular       test- 

0.00 

6.11 

0.32 

0.291 

4.9 

1  .79 

45.7 

H* 

channel  lined  with 

. 

.  .  . 

0.48 

0.417 

2.43 

53.8 

pebbles  1£   to   1$- 

.  .  . 

.  .  .• 

0.61 

0.510 

2.90 

58.0 

inch         diameter, 

.  .  . 

.  .  . 

0.73 

0.587 

3.27 

61  .1 

held  in  place  with 

.  .  . 

.  .  . 

0.84 

0  .636 

3.56 

62.8 

cement.  —  Darcy- 

0.93 

0.712 

3.85 

65.2 

Bazin.  Series  5. 

1  .03 

0  .772 

4.03 

65.5 

1  .13 

0  .823 

4.23 

66.6 

1.21 

0  .867 

4.43 

68.0 

1  .29 

0.909 

4.60 

69.0 

1.37 

0.946 

4.78 

70.3 

1.46 

0.987 

4.90 

70.4 

. 

XIII.  RECTANGULAR  TEST  CHANNELS  WITH  CLEATS  NAILED  CROSSWISE. 


Rectangular       test- 

0.41 

6.43 

0.33 

0.302 

1  .5 

1  .65 

77.4 

FA 

channel  of  boards, 

. 

0.51 

0.442 

2.17 

84.5 

with  cleats  1  inch 

0.89 

0.634 

2.86 

91  .0 

by  f  inch  nailed 

1.02 

0.775 

3.33 

94.0 

crosswise  on   bot- 

1 .23 

0.889 

3.68 

97.0 

tom   and  sides,  f 

, 

1.42 

0.986 

3.98 

99.0 

inch          apart.  — 

1.62 

1.076 

4.19 

99.0 

Darcy-  Bazin. 

Series  12. 

Do. 

6.43 

0.22 

0.205 

5.9 

2.50 

71  .8 

yrs 

Series  13. 

0.33 

0.302 

« 

3.34 

79.0 

0.51 

0.442 

tt 

4.40 

86.0 

0.67 

0.552 

(f 

5.08 

89.0 

0.80 

0.643 

(t 

5.63 

91.4 

0.92 

0.716 

tt 

6.14 

94.5 

... 

1  .05 

0.790 

a 

6.48 

94.8 

... 

Do. 

6.40 

0.19 

0.182 

8.9 

2.85 

70.8 

ydv 

Series  14. 

0.30 

0.273 

3.75 

76.4 

_ 

0.46 

0.403 

4.92 

82.4 

0.59 

0.499 

5.77 

86.8 

0.71 

0.582 

6.38 

88.9 

0.83 

0.658 

6.86 

89.9 

... 

0.94 

0.726 

n 

7.26 

90.5 

... 

Rectangular       test- 

0.03 

6.43 

0.43 

0.378 

1.5 

1.28 

53.7 

1 

7~ 

channel  of  boards 

0.66 

0.550 

ft 

1.68 

58.6 

7* 

with  cleats  nailed 

1.02 

0.777 

it 

2.21 

64.8 

crosswise  to    bot- 

1 .33 

0.942 

te 

2.55 

67.8 

tom     and      sides, 

1  .61 

1  .073 

tt 

2.81 

70.1 

cleats  1  by  f  inch, 

1  .91 

1  .197 

ft 

2.97 

70.0 

2  inches  apart.  — 

2.18 

1.299 

tt 

3.11 

70.5 

DarcyrBazin. 

.  .  . 

Series  15. 

102 


THE   FLOW   OF   WATER 


TABLE  X.  —  Continued. 


Description  of 
Conduit. 

m 

W 

d 

R 

1000s 

V 

c 

a 

Do. 
Series  16. 

... 

6.43 

0.29 

0.44 
0.67 
0.87 
1  .05 
1  .21 

0  .264 

0.384 
0.553 
0.686 
0.791 

0.882 

5.9 

1  .91 

2.56 
3.37 

3.88 
4.31 
4  .65 

48.3 

53.7 
59.0 
61  .0 
63.1 
64  .5 

1 
F* 

1.38 

0.965 

4.91 

65.1 

Do. 

Series  17. 

6.40 

0.25 

0.39 
0.60 
0.78 

0.252 

0.35 
0.501 
0.628 

8.86 

2.21 

2.85 
3.75 
4  .37 

48.7 

51.2 

55.8 
58.6 

1 
Fl 

... 

0.94 
1  .09 
1  .22 

0.725 
0.812 
0.885 

4.85 
5.22 
5.57 

60.5 
61.5 
62.9 

XIV.   CHANNELS  IN  ROCKWORK. 


Description  of 
Conduit. 

m 

K 

W 

d 

R 

1000s 

V 

c 

a 

Torlonia        Drain 
Tunnel,      Lake 
Fucino,     Italy. 
Section       oval, 
13.12  ft.     wide 
and    18.9       ft. 
high.          Total 
length,     20,666 
ft.,  of  which  §  is 
in         limestone 
rock,  the  rest  is 
lined  with  free- 
stone  masonry. 
—  Perrone,1894. 

-0.04 

1.07 

13.12 

... 

1  .93 
2.07 
2.55 
2.67 
3.24 
3.43 
3.52 
3.75 

1.04 

3.25 
3.62 
4.24 
4.32 
5.05 
4.95 
4.92 
5.35 

75.3 

76.2 
82.2 
81.7 
86.7 
82.9 
80.9 
86.2 

1.0 
1.0 
1  .027 
1  .013 
1  .027 
0.97 
0.94 
0.98 

Beacon         Street 
Tunnel,       Sud- 
bury  Aqueduct. 
Length  4592  ft. 
Bottom       lined 
with  rough  con- 
crete, sides  for 
the  greater  part 
unlined.       Sec- 
tion    oval.     — 
Fteley&  Steams 
1878. 

-0.02 

1.03 

10.0 

2.21 

0  .281 

1.97 

79.2 

1.0 

DESCRIPTION  OF   CONDUIT,    ETC. 


103 


TABLE  X.*—  Continued. 


Description  of 
Conduit. 

m 

K 

W 

d 

R 

1000s 

V 

c 

a 

Turlock          Rock 

-0.20 

1.50 

50.0 

10.0 

5.9 

1  .5 

7.5 

86,0 

1  .0 

Canal.       Exca- 

vated along  the 

banks    of    Tur- 

lock River,  Cal- 

ifornia. 

XV.   ARTIFICIAL  CHANNELS  IN  EARTH. 


Experiments  by 
S.  Fortier,  U.  S. 
Geol.  Survey, 
1901. 

Bear  River  Canal 
branch.  Well 
rounded  chan- 
nel in  clayey 
loam,  coated 
with  a  fine  sedi- 
ment. 

0.58 

0.27 

2.49 

0.31 

3.62 

130.2 

* 

Do. 
Providence  Canal. 

0.58 

0.27 

.  .  . 

.  .  . 

0.86 

0.12 

1.04 

102.6 

yh 

Do. 
Logan     &     Hyde 
Park  Canal. 

0.54 

0.27 

... 

... 

0.51 

1.88 

2.33 

95.3 

V* 

Channel  in  fine 
hard  gravel,  \ 
inch  in  diameter 

0.20 

0.66 

1.0 

0.32 

1  .44 

80.8 

yh 

Do. 

0.12 

0.77 

1.20 

0.83 

2.58 

81  .5 

yh 

Channel  in  clay 
with  fragments 
of  rock  \  inch 
in  diameter  im- 
bedded. 

0.07 

0.86 

•  •  • 

.  .  . 

1.07 

0.62 

1.94 

74.6 

yh 

Channel  in  sand 
some  plants  at 
edges. 

0.05 

0.90 

1.40 

0.15 

1.08 

75.4 

1  .0 

Channel  in  sand 
with  small  peb- 
bles imbedded. 

0.05 

0.90 

.  .  . 

0.52 

0.56 

1.01 

59.2 

1.0 

104 


THE   FLOW  OF   WATER 


TABLE  X.  —  Continued. 


Description  of 
Conduit. 

m 

K 

W 

d 

R 

1000s 

V 

c 

a 

Experiments  of  S 
Fortier  (contin- 

ued). 

Channel    in    sane 

--0.01 

1.04 

0.14 

1.35 

0.54 

39.3 

1  .0 

with  small  peb- 
bles imbedded. 

Channel     in     ce- 

0.0 

1.0 

1  .52 

0.77 

2.49 

73.0 

1.0 

mented    gravel 

1,2  and  3  inches 

in  diameter. 

Channel    in    sane 

-0.07 

1.17 

... 

... 

0.40 

0.40 

0.61 

48.0 

1.0 

with  gravel  im- 

bedded,          no 

vegetation. 

Do. 

-0.09 

1  .22 

... 

1.48 

0.43 

0.71 

64.9 

1.0 

Do. 

-0.08 

1  .18 

... 

... 

0.65 

0.75 

1.19 

54.0 

1  .0 

Do. 

-0.10 

1.24 

0.71 

1.75 

1.09 

53.8 

1.0 

Channel   in  earth 

-0.10 

1.24 

... 

0.55 

1  .16 

2.93 

50.4 

1.0 

with  gravel  im- 

bedded,   size    £ 

to  2  inches. 

Channel   in  earth 

-0  .15 

1.35 

... 

... 

0.65 

1.6 

1  .61 

50.8 

1.0 

with  fragments 
of  rock  imbed- 

ded, size  \  to  2 

inches. 

Channel  in  gravel 

-0.16 

1  .37 

1.62 

0.60 

1.94 

63.2 

1.0 

covered   with 

sediment,  grav- 

el    up     to     2£ 

inches  in  diam- 

eter. 

Channel  in  earth, 

-0.18 

1.48 

0.35 

1.3 

0.84 

39.8 

1.0 

bottom  covered 

with  fragments 

of  rock. 

Channel    in    cob- 
bles       covered 

-0.38 

2.52 

... 

0.52 

0.35 

0.43 

32.0 

1  .0 

with  silt,  edges 

irregular. 

DESCRIPTION   OF   CONDUIT,   ETC. 


105 


TABLE  X.  —  Continued. 


Description  of 
Conduit. 

m 

K 

W 

a 

R 

1000s 

V 

c 
25.9 

a 

Channel   in   loose 
gravel  up  to  1^ 
inches  in  diam- 
eter. 

-0.48 

2.44 

0.27 

9.91 

1.35 

1  .0 

Channel  in  earth, 
bottom         and 
sides         coated 
with   fragments 
of  rock  up  to  3 
inches  in  diam- 
eter. 

-0.42 

2.90 

0.20 

12.2 

1.02 

20.9 

1.0 

Rough  channel  in 
coarse       gravel 
and  cobbles. 

-0.40 

2.96 

0.23 

17.1 

1.33 

21.1 

1.0 

Do. 

-0.49 

3.62 

0.23 

17.0 

1.10 

17.9 

1.0 

Experiments     by 
Darcy-Bazin. 
Grosbois      Canal. 
Trape  zoidal 
channel        in 
earth.     No  vege- 
tation. Series  49. 

-0.12 

1  .28 

10.7 
11  .9 
14  .1 
15.7 

1  .4 
1  .9 
2.5 
2.9 

0-96 
1.32 

1.57 

1.78 

0.25 
0.275 
0.246 
0.275 

0.89 
1.34 
1.36 
1  .49 

57 
70 
69 
66 

1  .0 
0^98 

Do. 

Some   vegetation. 
Series  50. 

-0.33 

1.97 

10.5 
11.4 
13.8 
15.5 

1.5 

2.1 

2.7 
3.1 

1.05 
1  .42 
1  .65 

1.85 

0.31 
0.29 
0.33 
0.33 

0.82 
1  .26 
1  .30 
1.41 

45 
52 
56 

57 

1.0 

Do. 

Stony   earth,   lit- 
tle   vegetation. 
Series  37. 

-0.31 

1.89 

9.1 
11  .4 
12.6 
13.3 

1  .5 
2.0 
2.4 
2.7 

0.96 
.26 
.41 
.56 

0.792 
0.806 
0.858 
0.842 

1  .23 
1  .67 
1  .81 
2.0 

45 
53 
52 
55 

1.0 

Do. 
Series  41. 

-0.33 

1.97 

10.1 
12.0 
13.2 
14.3 

1.6 
2.3 
2.9 
3.0 

.04 
.38 
.57 
.71 

0.445 
0.450 
0.455 
0.441 

0.96 
.27 
.40 
.51 

45 
51 
52 
55 

1  .0 

Do. 
Covered      with 
vegetation      at 
many       points. 
Series  43. 

-0.37 

2.18 

10.1 
12.3 
13.5 
14.7 

1  .7 
2.4 
2.8 
3.1 

1  .06 
1  .41 
1  .60 
1.76 

0.42 
0.47 
0.43 
0.45 

.89 
.18 
.31 
.39 

42 
46 
49 
49 

1.0 

OF   THE 

UNIVERSITY 

OF 


106 


THE   FLOW   OF   WATER 


TABLE  X.  —  Continued. 


Description  of 
Conduit. 

m 

K 

W 

d 

R 

1000s 

V 

c 

a 

Do. 
Bottom  and  sides 

-0.38 

2.22 

9.9 
11  .1 

1.7 
2.2 

1.09 
1  .38 

0.464 
0.450 

0.82 
1  32 

42 
53 

1.0 

coated         with 
mud,       little 
vegetation.   Se- 
ries 47. 

12.5 
14.0 

2.7 
2.9 

1  .63 
1  .71 

0.479 
0.493 

1.43 
1  .68 

51 

58 

... 

Do. 

Section             wel 

-0.38 

2.22 

9.5 

10  7 

1.5 
2  1 

0.99 
1  30 

0.555 
0  555 

0.96 

1  48 

41 
55 

1  .0 

rounded      little 

11  9 

2  5 

1  56 

0  525 

1  57 

55 

vegetation     Se- 

13  5 

2  9 

1  71 

0  515 

1  75 

59 

ries  48. 

Do. 
Trapezoidal  chan- 

-0.44 

2.41 

10.5 
12  9 

1.9 
2  5 

1.14 
1  42 

0.678 
0  633 

0.91 

1  28 

33 
43 

i  6* 

nel     in     earth 

14  2 

2.9 

1  61 

0  644 

1  45 

45 

vegetation      at 

15  2 

3  2 

1  .74 

0.622 

1  65 

50 

many     points. 
Series  36. 

Chazilly       Canal. 
T  rapezoida] 
channel  in  stony 

-0.37 

2.18 

8.9 
11  .0 
12  1 

1  .5 
2.0 
2.4 

0.96 
1  .18 
1  .41 

0.957 
0.929 
0.993 

1  .24 
1  .70 
1  .80 

41 
51 

48 

1.0 

earth  ;     little 

13  .0 

2.6 

1  .54 

0.986 

1  .96 

50 

vegetation.   Se- 
ries 38. 

Canal  of  the  Jard. 
Channel           in 
earth  *  no  stones 

0.03 

0.95 

.  .  . 

1.68 
1.94 
2  05 

0  .0362 
0  .0362 
0  0458 

0.449 
0.479 
0  607 

57.6 
57.0 
62  6 

yh 

or       plants  — 

2  58 

0  0651 

1  069 

82  5 

I  0 

Dubuat,  1779. 

Millrace   of   Kag- 
iswyl         Very 

-0.07 

1.04 

1.04 
1  38 

1.754 
1  255 

2.819 
3  139 

65.8 
75  3 

FA 

regular  channel 

1  .41 

1  .20 

3  .221 

78  3 

in  earth,  bottom 
covered       with 
fine     gravel.  — 
Epper,  1885. 

Canal     at     Real- 
tore.         Trape- 
zoidal    channel 
in    earth  ;    bot- 
tom muddy.  — 
Darcy-Bazin. 

-0.18 

1.55 

19.7 

4.5 

2.87 

0.43 

2.54 

72.2 

1  .0 

DESCRIPTION  OF   CONDUIT,  ETC. 


107 


TABLE  X.  —  Continued. 


Description  of 
Conduit. 

m 

K 

W 

d 

R 

1000s 

V 

c 

a 

Ganges  Canal, 
Solani  Embank- 
ment. Trape- 
zoidal channel 
in  earth  ;  bed 
quite  uniform. 
—  A.  Cunning- 
ham. Series  197. 

-0.22 

1.54 

184.2 

9.7 

8.35 

0.22 

3.98 

92.8 

1.0 

Do. 
Bed        somewhat 
rough.       Series 
222-225. 

-0.22 

1.52 

64.0 
64.3 
64.8 
65.2 

4.6 
4.8 
5.1 
5.3 

4.07 
4.18 
4.37 
4.50 

0.306 
0.304 
0.297 
0.291 

2.71 
2.74 
2.79 

2.82 

76.8 
78.3 

77.4 
78.8 

1  .0 

Do. 

Bed  uneven.  Se- 
ries 192. 

-0.27 

1  .67 

174.9 

10.0 

8.64 

0.231 

3.98 

89.1 

1.0 

Canal         Cavour. 
Above    the   sy- 
phon     of     the 
Sesia.     Bottom 
width,  65.8  ft.; 
side  slopes,  1  :  1 
—  Passini    & 
Gioppi,  1892. 

-0.24 

1.60 

5.16 
5.83 
7.32 

0.29 
tt 

d 

3.10 
3.38 
3.70 

80.0 
80.2 
80.7 

1.0 

Do. 

Below  the  sy- 
phon of  the 
Sesia. 

-0  .28 

1.80 

... 

4.44 
5.25 
5.62 

0.33 
tt 

(S 

3.05 
3.08 
3.40 

72.2 

78.1 
79.0 

1.0 

Escher        Canal. 
Coarse      gravel 
and  detritus.  — 
Legler. 

-0.33 

1.94 

... 

... 

3.76 

4.42 

3.0 
tt 

6.986 
8.364 

65.7 
70.6 

1.0 

Linth  Canal  at 
Grynau.  Trape- 
zoidal channel 
in  earth  ;  bot- 
tom slightly 
rounded.  — 
Legler. 

-0.20 

1.50 

123 

5.14 
5.93 
6.48 
7.12 
7.52 
8.09 
8.28 
8.62 
8.87 
9.18 

0.29 
0.30 
0.31 
0.32 
0.33 
0.34 
0.34 
0.35 
0.36 
0.37 

3.41 
3.83 
4.15 
4.42 
4.72 
4.92 
5.06 
5.22 
5.39 
5  .53 

88.4 
90.8 
92.6 
92.6 
95.4 
93.8 
95.3 
95.1 
95.5 
94  9 

1.074 
1.052 

1  !6 
o!98 

108 


THE   FLOW  OF   WATER 


TABLE  X.  —  Continued. 


Description  of 
Conduit. 

m 

K 

W 

d 

R 

1000s 

V 

c 

a 

Do. 
At      Blaschen. 
Trapezoidal 
channel       in 
gravel  ;   bottom 
slightly    round- 
ed. 

-0.22 

1  .58 

113.2 

4.0 
6.5 

0.80 
0.41 

4.26 
4.17 

75.5 
80.7 

... 

Simme          Canal. 
Very         coarse 
gravel     and 
stones.  —  Wam- 
pfler,  1867. 

-0.44 

2.87 

... 

... 

1  .82 
1.87 
1  .36 
1.32 

6.5 
7.0 
11  .6 
17.0 

4.92 
5.37 
5.49 
5.99 

45.1 
46.9 
43.6 
39.8 

1.0 

Test    channel    in 
river     sand.  - 
Seddon,  1886. 

0.48 

0.57 

0.065 

0  .0465 

3.5 
3.7 
4.1 
5.2 

6.2 

0.86 
0.87 
0.84 
0.83 
0.86 

67.4 
67.8 
61.6 
54.0 
52.3 

1.0 

Do. 

0.48 

0.83 

0.040 

0  .0316 

7.9 
8.0 
8.6 
9.7 
11.3 

0.93 
0.89 
0.88 
0.91 
0.91 

58.9 
55.7 
53.2 
54.0 
51.1 

1  .0 

Triangular    chan- 
nel    in     sand  ; 
very  regular.  — 
Fresno. 

0.27 

0.57 

0.416 

0.312 

0.80 

71.0 

1  .0. 

Well     rounded 
irrigation  chan- 
nel    in     sandy 
soil.  —  Fresno. 

-0.03 

1  .04 

.  .  . 

1.0 

0.06 

0.50 

64.5 

1.0 

XVI.   ARTIFICIAL  CHANNELS  IN  EARTH  WITH  SIDE  WALLS  OF  MASONRY. 


Millrace  at  Pris- 
bram.  Very 
regular  channel 
in  clay  with 
side  walls  of 
masonry.  Sec- 
tion trapezoid- 
al. —  Rittinger, 
1855. 

0.12 

0.79 

0.54 
0.66 

0.373 
0.425 

1  .0 

u 

1.127 
1.254 

58.4 
60.8 

1.0 

Do. 

Trapezoidal  chan- 
nel in  earth 
with  dry  rubble 
side  walls.  Very 
irregular. 

-0.62 
-6.43 

5.09 
2'.i6 

... 

0.41 
0.44 
0.70 
0.80 
0.86 
0.90 

0.316 
0.336 
0.472 
0.548 
0.560 
0.566 

2.2 

i 

t 

c 
( 
{ 

0.289 
0.588 
0.953 
1.135 
1  .190 
1.269 

14.8 
21.6 
29.6 
32.7 
33.9 
36.0 

1  .0 
1  .0* 

DESCRIPTION  OF   CONDUIT,    ETC. 

TABLE  X.  —  Continued. 


109 


Description  of 
Conduits. 

m 

K 

W 

d 

R 

1000s 

V 

c 

a 

Millrace  at  Bezy- 
^banya.     Trape- 
zoidal    channe 
in     sand     a  n  c 
gravel  with  side 
walls  of  mason- 
ry. —  Rittinger 

-0.42 

6  '02 

2.79 
0'.74 

... 

0.28 
0.35 
0.56 
0.73 
0.90 

0.242 
0.282 
0.407 
0.483 
0.561 

5.0 
n 

d 
d 
a 

0.782 
1  .191 
1  .956 
2.134 
3  .475 

22.5 
31.7 
43.4 
43.4 
65.6 

1.0 

i.o' 

Millrace    at    Dio- 
sgyor.    Rectan- 
gular channel  in 
clay    with    side 
walls      of     dry 
rubble  masonry 
—  Rittinger. 

0.07 

-0^28 

1  .0 

l'.38 

0.63 
1  .14 
1.66 

0.487 
0.736 
0.924 

4.0 

2.463 
2.750 
3.323 

54.5 
50.7 

54.8 

1.0 

i  !o 

Embankment     oi 
Solani     Ganges 
Canal,       India. 
Main  Site. 
Built  up  channel 
in    clayey    soil 
with  many  arti- 
ficial    bars     of 
masonry       and 
boulders.     Side 
walls     of     ma- 
sonry   built    in 
steps,  bed  very 
rough,  masonry 
damaged         in 
places.  The  first 
four  gagings  in- 
dicate the  pre- 
pondering  influ- 
ence     of      the 
great  roughness 
of  the  bed.   — 
Allan  Cunning- 
ham. 

150.0 

it 

a 
a 

152.3 

157.0 
159.3 
161  .3 
164.0 
166.3 
168.7 
170.1 

1  .7 
2.3 
3.9 

4.1 
5.6 

6.8 
7.6 
8.2 
9.1 
9.9 
10.7 
11  .0 

1  .69 
2.26 
3.86 

4.07 
5.39 

6.18 
6.78 
7.26 
7.84 
8.42 
8.96 
9.34 

0.090 
0.148 
0.088 

0.215 
0.155 

0.171 
0.221 
0.214 
0.215 

0.227 
tt 

ii 

0.44 
0.87 
1.35 

1.79 
2.40 

3.05 
3.39 
3.22 
3.43 
3.58 
3.71 
4.02 

35.7 
45.9 
73.2 

66.5 
83.0 

93.8 

87.5 
81.7 
83.6 
83.6 
82.3 
87.3 

*i 

-0.20 

1.37 

yA 

Do. 
Jaoli     Site. 
Side  walls  lined 
with    brick    set 
in     clay  ;      side 
slopes  1  :  2. 

-0.17 

1.32 

190.9 

191  .2 
191  .5 

191  .8 
192.3 

6.8 

7.0 
7.3 

7.6 

8.1 

6.32 

6.53 
6.79 
7.05 
7.46 

0.140 

0.144 
0.145 
0.146 
0.166 

2.63 

2.76 
2.80 
2.81 
2.94 

88.4 

88.1 
89.2 
87.6 
85.1 

1 

Vh 

... 

110 


THE    FLOW  OF  WATER 


TABLE  X.  —  Continued. 


Description  of 
Conduit. 

m 

K 

W 

d 

R 

1000s 

r 

c 

a 

8f» 

OOAC 

3(\7 

ne     A 

1 

Do. 
Belra  Site. 

-0  .45 

.97 

187  .3 

187.5 

188.0 

.0 

8.7 
9.5 

.yt> 

8.21 

8.72 

.ZUo 

0.198 
0.200 

.(Jl 

3.01 
3.12 

to  .4 

74.7 

74.7 

FrV 

188.4 

9.6 

9.02 

0.191 

3.17 

76.4 

XVII.   NATURAL  CHANNELS  IN  EARTH. 


Description  of 

K 

W 

d 

R 

1000s 

r 

c 

a 

Channel. 

1 

La      Plata      River, 

1  .03 

16.22 

0.007 

1.391 

128.3 

i 

Catalina    channel. 

FT8 

Width  of  channel 

many  miles.     Bed 

fine  sand.     Slopes 

measured        with 

great  accuracy  for 

a   distance    of   85 

miles.—  J.J.  Re  vy. 

1 

Parana  de  las   Pal- 

1.12 

1222 

50.3 

0.007 

3.07 

160.0 

r~ 

mas. 

71* 

Do. 

. 

49.7 

0  .0068 

2.95 

160.3 

... 

49.5 

" 

2.87 

156.6 

1 

Parana.          Rosario 

1.18 

2460 

44.6 

0  .0058 

2.63 

152.9 

_ 

Section. 

v^ 

Do. 

1 

Seine  River  at  Paris. 
Section      between 

1  .45 

9.48 

0.14 

3.37 

92.^ 

vs 

the  bridges  of  Jena 

10.92 

K 

3.74 

95.6 

and    "The    Inva- 

12.19 

" 

3.80 

92.4 

lides."—  Ville- 

. 

14.50 

(( 

4.23 

94.0 

'.  .  '. 

vert. 

15.02 

(( 

4.51 

98.3 

15.93 

0.173 

4.68 

89.5 

16.85 

0.131 

4.80 

102.1 

... 

... 

... 

18.39 

0.103 

4.69 

107.6 

1    Qfi 

9C 

01  1  Q 

9    RO 

00      C 

I 

JtxlVCr     JL  O     £lt/    Jj  OSS3. 

d'Albero.        Com- 

i .yo 

... 

.0 

.i±y 

£  .DU 

oo  .O 

v& 

mission  of  Italian 

10.1 

0.12 

2.92 

88.1 

Engineers,  1878. 

11.0 

0.12 

3.15 

112.0 

9.8 

0.103 

3.25 

96.0 

11  .8 

0.104 

3.28 

100.0 

Do.     At    Porto   Mo- 

. 

. 

12.4 

0.093 

3.37 

95.0 

.  '.  '. 

rone. 

.  .  . 

.  .  . 

.  .  . 

8.5 

0.165 

3.04 

81  .5 

DESCRIPTION  OF    CONDUIT,  ETC. 


Ill 


TABLE  X. — Continued. 


Description  of 
Channel. 

K 

W 

d 

R 

1000s 

X 

c 

a 

Weser     River    near 

1  .92 

9.4 

0  .2499 

4.07 

83.8 

1.0 

Minden. 

9.82 

tt 

4.37 

87.7 

—  Schwartz,  1808. 

10.57 

tt 

4.75 

92.8 

11  .18 

" 

4.94 

93.2 

12.07 

tt 

5.24 

95.0 

12.64 

" 

5.26 

94.2 

12.66 

tt 

5.35 

97.0 

... 

... 

14.13 

ec 

5.67 

96.0 

0^98 

Do.  near  Minden. 

1  .98 

4.50 

0  .5032 

3.38 

71.0 

1  .0 

5.33 

tt 

4.0 

78.6 

6.14 

tt 

4.88 

88.0 

6.66 

" 

5.24 

90.5 

7.59 

" 

5.75 

93.2 

8.14 

" 

5.95 

93.2 

8.61 

" 

6.15 

93.2 

10.23 

tt 

6.66 

93.2 

10.45 

tt 

6.56 

90.6 

10.71 

" 

6.56 

94.0 

... 

11.28 

ei 

6.94 

92.0 

I'oi 

T")n        At  Vlrvtnw 

I       K1 

A       Off 

OKKfJO 

Ol      f\ 

1 

UO.      jt\.\j    VlOtOW. 

1    .Ol 

7.41 

.oouo 
tt 

5.32 

ol  .U 

84.0 

FA 

8.92 

" 

6.30 

90.0 

9.3 

tt 

6.52 

91.2 

9.72 

tt 

6.65 

91.2 

11  .70 

If 

7.53 

93.8 

13.0 

tt 

7.92 

93.9 

13.35 

tt 

7.90 

92.1 

Pless^ir  River.  Some 

1  .55 

1  .25 

9.65 

6.6 

54.7 

1.0 

stones  and  gravel. 

2.33 

it 

9.99 

66.4 

—  La  Nicca,  1839. 

. 

3.48 

" 

10.19 

66.4 

3  .58 

" 

13.58 

72.4 

... 

3.59 

M 

13.94 

74.8 

Saalach  River.  Some 

1.61 

1.54 

0.875 

2.073 

56.5 

1  .0 

stones.  —  Roff, 

1.31 

1.100 

2.246 

58.8 

1854. 

1.91 

1  .242 

3.077 

63.0 

1  .98 

1.240 

3.385 

68.2 

... 

2.16 

3.660 

5.474 

64.3 

River  Rhine  in  Dom- 

1.71 

0.25 

5.74 

1.25 

32.8 

1 

leschger      Valley. 
Gravel  and  detri- 

1.32 

7.73 

4.75 

47.0 

F" 

tus.  —  La    Nicca. 

2.95 

7.95 

7.42 

48.3 

.  .  . 

112 


THE   FLOW  OF   WATER 


TABLE  X.  —  Concluded. 


Description  of 

K 

W 

d 

R 

1000s 

r 

c 

a 

Channel. 

1 

Salzach  River.  From 

1.94 

3.53 

0.94 

3.48 

60.3 

yh 

Bergheim      to 

4.20 

0.94 

4.03 

63.9 

Wildshut.    Gravel 

7.39 

1.12 

5.786 

63.4 

and       detritus.  — 

3.51 

1.55 

4.10 

55.4 

Reich.    • 

4.64 

1.55 

4.67 

67.5 

3.87 

1.79 

4.45 

53.4 

4.26 

1.79 

5.15 

58.8 

... 

Zihl  River  near  Gott- 

1.98 

3.52 

0.4 

2.296 

61.0 

1 

statt.      Bed    very 

"j/rir 

irregular,   covered 

5.02 

" 

3.706 

77.1 

with  mud  and  de- 

5.53 

" 

4.625 

69.1 

tritus.  —  Trechsel, 

1825. 

Mississippi    River  at 

414 

Q100 

64   Q 

Orjo 

91  9 

1 

Columbus,  Ky.,  at 

^t    .-LTT 

O  L44 

vT±    .  t7 

.Uo 

' 

y& 

high  water,  1895. 

Rep.       of      Miss. 

River     Com.     of 

1896. 

Do.  At  Helena,  Ark. 

2.91 

5100 

40.5 

0.07 

5.207 

97.8 

Do.  At  Arkansas  City 
Do.  At  Wilson  Point. 

4.31 
2.68 

3453 
3944 

... 

65.2 
56.4 

0.064 
0.054 

5.807 
6.145 

89.7 
111  .4 

(C 

La. 

Do.        At  Natchez, 

1  .13 

2173 

69.5 

0  .0459 

9.512 

161  .2 

(t 

Miss. 

Do.      At  Red  River 

1  .59 

4044 

57.2 

0  .0284 

5.636 

140.9 

tl 

Landing,  La. 

Do.     At  Carrollton, 

1.22 

2338 

71  .0 

0  .0219 

6.494 

162.3 

it 

La.,  at  high  water. 

72.7 

0  .0254 

6.254 

161.8 

Bed  sand. 

Do.     At  low  water. 

1  .51 

2338 

65.5 

0  .0021 

1.842 

158.0 

" 

1 

Irrawaddi    River  at 

2.66 

3395 

35 

16.28 

0  .00861  1  .007 

85  .1      1  .0 

Saiktha,  Burmah. 

39 

18.49 

0  .0172 

1  .783 

99.9 

Bed      sand     with 

43 

19.99 

0  .0218 

2.360 

103.9 

stones,  right  bank 

47 

21.13 

0  .0344 

2.857 

105.9 

1  17 

rocky  in  places.  — 

53 

26.42 

0  .0474 

3.548 

100.3 

Gordon,  1873. 

. 

57 

29.80 

0  .0559 

3.993 

97.8 

.  .  . 

63 

35.44 

0  .0688 

4.052 

94.2 

.  .  . 

. 

69 

41  .01 

0  .0817 

5.382 

92.9 

73 

44.47 

0  .0904 

6.147 

97.0 

0  .84 

FOKMS   OF  SECTIONS   OF  CONDUITS  113 

Forms  of  Sections  of  Conduits. 

RELATION  OF  MEAN  HYDRAULIC  RADIUS  TO  WET  PERIMETER. 

In  the  design  of  the  form  of  cross-section  of  an  artificial  con- 
duit two  factors  enter  into  consideration: 

1.  The  material  composing  the  walls  of  the  channel. 

2.  The  special  purpose  for  which  it  is  intended. 

Conduits  under  pressure,  whether  constructed  of  metal, 
wood,  earthenware,  concrete  or  masonry  are  nearly  always 
circular  in  section,  because  this  form  can  best  be  given  the 
strength  to  resist  internal  and  external  pressures.  The  thickness 
of  the  material  forming  the  walls  of  a  circular  conduit  is  found 

from  the  formula: 

PD   , 
<  =  -jr  +  c 

in  which  P  is  the  pressure  in  pounds  per  square  inch; 
D  the  internal  diameter  in  inches; 
T  the  safe  tensile  strength  of  the  material; 
c  a  constant  added  to  guard  against  defects  in  the 
casting  or  the  welding. 

For  such  conduits  as  are  subject  to  water  ram  a  pressure  of 
100  pounds  per  square  inch  is  allowed  in  addition  to  the  pressure 
due  to  the  head  which  is  equal  to  P  =  0.434  h.  The  stresses 
allowed  in  the  material  and  the  constants  added  are : 

For  cast  iron  T  =    4,000,  c  =  0.33 

For  wrought  iron  T  =  17,000,  c  =  0.06 

For  steel  .  T  =  20,000 

For  lead  T  =       450,  c  =    0.3 

For  concrete,  2  per  cent  steel  T  =       480,  c  —    1.0 

Since  the  advent  of  reinforced  concrete,  conduits  constructed 
of  this  material  are  coming  more  and  more  into  favor.  Steel- 
concrete  water  pipes  resisting  pressures  of  heads  exceeding  100 
feet  are  now  in  use.  The  two  aqueduct-syphons  of  Sosa  have 
internal  diameters  of  12.46  feet,  and  resist  the  pressure  of  a 
head  of  92  feet. 


114 


THE    FLOW   OF  WATER 


FIG.  4. 
Forms  of  Sections  of  Masonry  Conduits.     The  Numbers  are  Proportional. 


FORMS  OF  SECTIONS  OF  CONDUITS 


115 


Steel  concrete  sewer  pipe  is  now  made  in  diameters  from  15  to 
120  inches. 

Open  conduits  lined  with  concrete  are  most  frequently  made 
semicircular  in  section.  Wooden  flumes  which  are  acting  simply 
as  aqueducts  are  generally  made  semi-square  in  section,  if  they 
are  intended  to  carry  lumber  or  wood  the  triangular  section  is 
used.  For  aqueducts  lined  with  masonry  a  section  is  generally 
preferred  whose  sides  are  vertical  or  nearly  so,  whose  bottom  is 
a  flat  segmental  arch,  and  whose  top  (if  covered)  is  a  semi- 
circle. Very  large  aqueducts,  those  crossing  valleys,  streams 
or  other  depressions  are  given  a  rectangular  section. 

In  designing  channels  in  earth  the  velocity  enters  into  the 
problem.  In  those  of  some  dimensions  the  bottoms  are  well 
rounded  and  the  sides  given  slopes  ranging  from  J  to  1  for 
cemented  gravel,  to  3  to  1  for  loose  sand. 

In  a  preceding  chapter  we  have  observed,  that  the  form  of  the 
cross-section  of  a  conduit  has  an  appreciable  influence  on  the 
power  of  the  velocity  to  which  the  frictional  resistance  is  pro- 
portional and  that  the  circular  or  semicircular  form  is  the  one 
most  favorable  to  flow.  For  rectangular  conduits  lined  with 
boards,  for  instance,  we  have  found  the  value  of  the  coefficient 
a  to  be  equal  to  V&  and  equal  to  W*  for  semicircular  conduits 
lined  with  the  same  material.  The  circular  form  has  the 
additional  advantage  of  having  a  wet  perimeter  less  in  propor- 
tion to  the  area  of  the  section  than  any  other  form. 
>A  F  E  D  Let  AD  (Fig.  5)  be 

the  top  width  of  a  trap- 
ezoidal channel,  BC  its 
bottom  width,  and  FB 
the  depth. 

The  area  of  the  cross- 
section  will  then  be : 
y 

FIG.  5. 


116  THE  FLOW   OF    WATER 

and  the  wet  circumference 

P  =  AB  +  BC  +  C  D. 
Let  •     BC  =  6, 

BF  =  d, 

AD  =  t, 

AF_ 

BF 

Then  the  area  A  =  db  +  Id2  =  d  (b  +  Id); 

the  wet  perimeter          P  =  b  +  2  d  \/l  +  Z2; 

the  bottom  width  b  =    -=-  —  Id; 

d 
p 
and  the  relation  — ,    the  reciprocal  of  R, 

P       1       d 

-T  =  2  +  -T  (2  Vl*  +  !  -  0  ' 

-il  Cc-  ./I 

Let  the  angle  which  the  side  of  the  conduit  CD  makes  with 
the  horizontal  be  denoted  by  a,  and  we  have  for  the  conditions 

P  A 

most  favorable  to  flow  or  — T-  a  miminum,  and  —  a  maximum, 

A  sr 


the  depth  d  =  if _ 


A  sin  a 


2  -  cos  a' 
the  top  width  t  =  b  +.2ld  =  —  +  d  cotangent  a; 

A 

the  bottom  width  b  =  -r  -  d  cotangent  a; 
a 

, .       ,  , .       Wet  Perimeter 
the  relation 


Area  of  Section 

jP  __6_        2rf 
^4.       ^       sin  a 


=  l  +  /2-cosa)\d 
d      \    A  sin  a   I 


Consequently,  for  a  given  value  of  i?  and  given  side  slopes  the 
area  of  section  is  least  if  R  is  equal  to  one-half  the  actual  depth  of 
water. 

For  the  semicircular  section  R  is  equal  to  one- half  the  radius, 
hence  this  forms  fulfils  the  conditions  best  and  other  forms  of 


SEWERS 


117 


section  fulfil  it  the  better  the  nearer  they  approach  the  semi- 
circle. Table  A  contains  values  of  R  and  areas  of  sections  in 
terms  of  the  radius  or  semi-diameter  *for  semicircular  conduits. 

TABLE  A. 


Depth  of 
Water  in 
Terms  of 
Radius. 

Value  of  R 
in  Terms  of 
Radius. 

Wetted  Sec- 
tion in  Terms 
of  Radius. 

Depth  of 
Water  in 
Terms  of 
Radius. 

Value  of  R 
in  Terms  of 
Radius. 

Wetted  Sec- 
tion in  Terms 
of  Radius. 

0.05 

0.0321 

0.0211 

0.55 

0.320 

0.709 

0.10 

0.0524 

0.0598 

0.60 

0.343 

0.795 

0.15 

0.0963 

0.1067 

0.65 

0.365 

0.885 

0.20 

0.1278 

0.1651 

0.70 

0.387 

0.979 

0.25 

0.1574 

0.228 

0.75 

0.408 

1.075 

0.30 

0.1852 

0.298 

0.80 

0.429 

1.175 

0.35 

0.2142 

0.370 

0.85 

0.439 

1.276 

0.40 

0.2424 

0.450 

0.90 

0.446 

1.371 

0.45 

0.2690 

0.530 

0.95 

0.484 

1.470 

0.50 

0.2930 

0.614 

1.00 

0.500 

1.571 

Table  B  contains  proportions  of  channels  of  maximum  values 
of  R,  the  mean  hydraulic  radius  for  a  given  area  and  given 
side  slopes. 

Half  the  top  width  is  the  length  of  each  side  slope  and  the  wet 
perimeter  is  the  sum  of  the  top  and  bottom  widths.  The  mean 
hydraulic  radius  is  equal  to  one-half  the  depth  of  water. 

TABLE  B. 


Description  of  Form  of 
Section. 

Inclination  of 
Sides  to 
Horizon. 

Ratio  of 
Side 
Slopes. 

Area  of 
Section  in 
Terms  of 
Depth. 

Bottom 
Width  in 
Terms  of 
Depth  of 
Water. 

Top 

Width  in 
Terms  of 
Depth  of 
Water. 

Semi  Circle 

571  d2 

Semi  Hexagon  .    .    . 
Semi  Square  .... 
Trapezoid    
Do  
Do  

60° 
90° 
78°  58' 
63°  26' 
53°  8' 

3     5 
0     1 
1     4 
1     2 
3     4 

.732  d2 
2d2 
.812d2 
.736d2 
.750d2 

1.155d 
2d 
1  .  562  d 
1  .  236  d 
d 

2.31d 
2d 
2.062d 
2.236d 
2.50d 

Do  

45.0 

1 

1.828d2 

0.828d 

2.828d 

Do 

38°  40' 

U 

1  952  d2 

0  702  d 

3  022  d 

Do  
Do 

33°  42' 
29°  44' 

1* 
if 

2.106d2 
2  282  d2 

0.606d 
0  532  d 

3.606d 
4  032  d 

Do  
Do         ... 

26°  34' 

23°  58' 

2 

2i 

2.472d2 
2  674  d2 

0.472d 
0  424  d 

4.472d 
4  924  d 

Do. 

21°  48' 

2i     1 

2  885  d2 

0  385  d 

5  385  d 

Do.    .    .    . 

19°  58' 

2|     1 

3  104  d2 

0  354  d 

5  854  d 

Do  

18°  26' 

3     1 

3  325  d2 

0  325  d 

6  325  d 

118 


THE    FLOW   OF  WATER 


Sewers. 

The  forms  of  the  cross-section  most  frequently  adopted  for 
sewers  are  the  circular  and  the  oval  or  egg-shaped.  Only  sewers 
of  very  great  dimensions  are  given  a  rectangular  section,  the 
roof  being  a  flat  segmental  arch. 

For  sewers  less  than  two  feet  in  diameter  glazed  earthenware 
pipe  is  mostly  used,  less  frequently  concrete  pipe.  Sewers  con- 
structed of  brick,  masonry  or  concrete  are,  however,  found  with 
diameters  down  to  two  feet. 

When  the  discharge  of  a  sewer  is  estimated  to  be  fairly  con- 
stant the  circular  section  is  preferred,  when  it  varies  con- 
siderably, however,  some  form  of  an  oval  sewer  is  used. 


Two  forms  of  egg-shaped  sewers  are  in  general  use.  The  one 
most  frequently  adopted  has  the  proportional  parts  as  given  in 
the  annexed  figure.  In  the  other  form  the  lower  circle  has  a 
diameter  of  one-fourth  the  diameter  of  the  upper  circle  only. 
This  form  is  used  for  sewers  of  small  dimensions  and  greatly 
varying  in  discharges. 

The  vertical  diameter  in  both  forms  is  always  equal  to  1J 
diameters  of  the  upper  circle. 


SEWERS  119 


In  the  figure         Tangent  —  =       =  0.75. 


Hence  the  angle  BCG  =  36°  53', 

and  the  angle          EGG  -  180°  -  (90°  +  36°  53')  =  53°  7', 
hence,  the  angle      FGD  =  2  X  53°  7'  =  106°  14'. 

Using  trigonometry,  these  data  enable  us  to  compute  the 
different  parts  of  the  area  and  the  circumference  with  precision. 

For  the  proportional  parts  given  in  the  figure  we  find  by  this 
method  : 

The  area  =  18.35  which  is  equal  to  1.147  d2, 

the  circumference  =  15.8488  which  is  equal  to  3.9622  d, 

the  mean  hydraulic  radius  =  1.1584,  which  is  equal  to  0.2896  d, 
d  being  the  horizontal  diameter. 

By  the  same  method  we  may  compute  the  value  of  the  mean 
hydraulic  radius  or  the  area  for  any  depth  of  water  in  the  sewer. 
The  mean  hydraulic  radius  has  its  greatest  value  when  the  con- 
duit is  about  0.85  full.  It  is  equal,  being 

0.85  full  to  0.345  the  horizontal  diameter; 
|  full  to  0.33  the  horizontal  diameter; 
i  full  to  0.28  the  horizontal  diameter; 
J  full  to  0.20  the  horizontal  diameter. 

The  speed  of  flow  necessary  to  prevent  a  deposit  of  sewage  is 
given,  for  all  forms  of  the  cross-section  by  the  equation 


or,  in  exceptional  cases,  when  the  sewer  is  very  well  constructed 

by 

0.0625 


0 

v  =  2  H  , 


r  being  the  mean  hydraulic  radius. 

Hence  for  the  very  greatest  section  the  least  permissible 
velocity  is  two  feet  per  second.  In  order  that  the  velocity 
should  not  fall  below  the  permissible  limit  the  value  of 

66  (  Vr  +  ™)  Vr, 


120  THE   FLOW   OF   WATER 

on  which  for  equal  slopes  the  velocity  depends  should  not,  for 
any  quantity  of  discharge  vary  greatly. 

If  we  assume  the  horizontal  diameter  to  be  equal  to  4  feet, 
the  value  of  r  is,  for  the  sewer  running  full,  equal  to  1.1584  feet, 
for  the  sewer  running  \  full  to  0.80  feet. 

Taking  m  =  0.57  (corresponding  to  common  brickwork)  the 
value  of  66  ( Vr  +  m)  A/r  is  in  the  first  case  equal  to  113.6,  in  the 
second  to  89.48. 

For  r  =  0.8  the  velocity  necessary  to  prevent  a  deposit  is 
equal  to 


2  +  ^^  =  2.156  ft.  per  second. 
0.8 


s  =     7~v=T  T~7=l  =  R^1  I  =  °00053 

L66  (  Vr  +  m}  V r  J       L  8r 


For  this1  velocity  the  slope  will  be 

.156m2 

(  Vr  +  m)  Vr  J   ~  L  89.48  J 
For  the  same  slope  and  the  sewer  running  full  the  velocity  will 
be 

v  =  (66  ( Vr  +  m)  Vrs)&  =  (113.6  X  0.02304)if  =  2.575  feet 
per  second. 

The  cross-section  is  equal  to  1.147cP  =  18.864 /2,  hence  the 
discharge,  for  the  sewer  running  full 

Q  =  18.864  X  2.575  =  48.57  /3 


and  Q  =  x  2.516  =  9.09  /3 

for  the  sewer  running  \  full.  Thus,  while  the  actual  discharge  in 
cubic  feet  per  second  for  the  sewer  running  full  is  5.34  times  the 
discharge  of  the  sewer  running  J  full,  the  difference  in  the  speed 
of  flow  is  only  2.575  —  2.156  =  0.419  feet  per  second. 


EXPONENTIAL   EQUATIONS  121 

EXPONENTIAL  EQUATIONS. 

General  Relations  between  Diameters  and  Velocities  or  Quan- 
tities. General  Relations  between  Slopes  and  Velocities  or 

Quantities. 

Long  Circular  Conduits  Running  Full. 

A. 

If  in  our  general  equation  for  the  velocity  of  flow  we  substitute 
d,  the  diameter  of  a  conduit  in  feet,  for  r,  its  mean  hydraulic 
radius,  the  formula  thus  transformed  will  read: 
v  =  23.34  ( V5  +  1.414  ra)  Vds 

and  for  a  =  vv 

v  =  (23.34  (V5  +  1.414m)  V~ds)$ 
which  may  be  written 

v  =  34.607  (  V5  +  1.414  m)*  d&  s*. 

For  the  term  ( *V3  +  1.414  m)  and  its  variation  with  the 
velocity  we  have  no  adequate  substitute,  containing  as  it  does 
two  variables  in  an  everchanging  relation.  This  fact  makes 
the  problem  of  finding  an  exponential  equation,  giving  values  as 
exact  as  those  found  from  the  general  formula  an  impossibility. 
The  powers  of  the  diameter  (or  the  mean  hydraulic  radius)  to 
which  velocities  and  quantities  are  proportional  are  not  con- 
stant, even  for  the  same  degree  of  roughness,  but  vary  with  the 
diameter  (or  the  mean  hydraulic  radius)  itself.  On  this  account 
exponential  equations  with  constant  values  of  the  powers  of 
d  or  r  are  only  approximations,  fairly  true  between  certain 
limits,  but  the  more  incorrect  the  farther  outside  of  these  limits. 
Such  equations  should  only  be  considered  as  brief  empirical 
expressions,  valuable  only  on  account  of  their  brevity;  they 
should  never  be  substituted  for  the  general  formula  when  a 
great  degree  of  accuracy  is  desired. 

Computing  the  velocities  and  discharges  of  two  long  straight 
circular  conduits  of  different  diameters,  but  of  the  same  degree  of 
roughness  and  having  the  same  slope  from  the  general  equation, 
we  may  by  means  of  the  data  thus  obtained  find  an  expression  for 
the  relation  between  the  diameter  and  the  velocity  or  the  discharge 
which  holds  good  between  the  limits  of  the  two  values  of  d. 


122  THE   FLOW   OF   WATER 

To  find  the  exponents  of  the  powers  of  d  to  which  velocities 
and  quantities  are  proportional  we  may  put 

x  _  log  V,  -  log  VQ 

log  dl     --log  dQ 

y  _  log  Qt  -  log  Q0 

By  means  of  these  equations  and  for  values  of  d  between  1 
and  50  inches  for  a  =  v\,  and  between  1  foot  and  20  feet  for 
other  values  of  a  we  find  the  following  values  of  x  and  y. 

a  =  v\  m  =  1.0  x  =  0.67  y  =  2.67 

a  =  vl  m  =  0.95  x  =  0.67  y  =  2.67 

a  =  v\  m  =  0.83  x  =  0.68  y  =  2.68 

a  =  v?  m  =  0.68  x  =  0.695  y  =  2.695 

a  =  V&  m  =  0.57  x  =  0.70  y  =  2.70 

a  =  m\  m  =  0.53  x  =  0.70  y  =  2.70 

a  =  i.o  m  =  0.57  x  =  0.66  y  =  2.66 

a  =  1.0  m  =  0.45  x  =  0.67  y  =  2-67 

a  =  i.o  m  =  0.30  z  =  0.68  y  =  2.68. 

Consequently,  for  m  =  0.68  (pipes  of  planed  staves,  cast  and 
wrought  iron,  etc.,  all  some  time  in  use),  velocities  are  pro- 
portional to  d0'695,  quantities  to  d2'695  and  we  have  between  V 

and  d  the  relation 

/7  °-695 

• (1) 


J  0.695 
1 


^  0-695 
'1=^03^695- («) 

a0 

1.453 

,,-   -  <» 

and  between  Q  and  d 

Q,  _  ^  r2) 

f}      ~~    J    2-695  W 

^o       ao 

rfi\2'695  /  ^ 

(a) 

(6) 


EXPONENTIAL  EQUATIONS 


123 


Equations  (a)  1  and  2  enable  us  to  find  velocities  and 
quantities  for  a  diameter  dlf  provided  velocities  and  quantities 
for  a  diameter  d0  are  known. 

From  equations  (6)1  and  2  we  may  find  the  diameter  dt  for 
a  velocity  Vl  or  a  quantity  Qv  provided  the  diameter  d0  for  the 
Velocity  VQ  or  the  discharge  Q0  is  known. 

In  the  same  manner  we  find  for  the  relation  between  the  slope 
and  velocities  and  quantities: 


(3) 


Co        «  * 


(4) 
(  } 


» 


Combining  equation  (1)  and  (3)  we  have: 


124 


THE    FLOW   OF   WATER 


Combining  (2)  and  (4)  we  have: 


(6) 


(a) 


=  <* 


(c) 


By  means  of  these  equations  we  may  find : 

The  value  of  V1  or  Q1  for  any  value  of  dl  or  $r 

The  value  of  d^  for  any  value  of  7l;  Ql;  or  $r 

The  value  of  St  for  any  value  of  Vlf  Q±  or  d1 
provided  values  of  F07  Q0,  d0,  S0  are  known.     In  these  equations 
S&  is  substituted  for  S^5  when  a  =  v&  and  <S*  when  a  =  1.0. 

Computing  velocities  and  quantities  of  discharge  for  a  con- 
duit one  foot  in  diameter  and  having  different  degrees  of  rough- 
ness from  the  general  formula  we  find  for  three  values  of  (a) 
the  following  equations: 

1.  For  a  =  v*, 

v  =  93.25  -  59.84  (1  -  m)  d* S&, 
Q  =  76.69  -47.0  (l-m)d»&*. 

2.  For  a  =  ^, 

v  =  70.44  -  41.85  (1  -  m)  d*  S&, 
Q  =  53.33  -  32.87  (1  -  m)  ^  S&. 

3.  For  a  =  1.0, 

v  =  56.33  -  33.0     (1  -  m)  d*  S*t 
Q  =44.24  -25.92  (1-m)  ^5*. 
In  particular  we  have  the  following: 

m  =  1.0.       New  long  straight  conduits  lined  with  clean  cement 
very  smooth.    Tin  pipes.    Plated  pipes. 
v  =93.25  d°-wSA, 
=  76.69  da- 


EXPONENTIAL   EQUATIONS  125 

m  =  0.95.     Very  smooth  new  asphalt-coated  cast  or  wrought- 
iron  pipes,  also  new  asphalt-coated  wrought-iron 
and  steel  riveted  pipes  not  exceeding  6"  in  diame- 
ter.    New  conduits  of  planed  staves. 
v  =  90.17  d°-e*S&, 
Q  =  70.82  da-fl8S*. 

m  =  0.83.  Ordinary  new  asphalt- coated  cast  and  wrought-iron 
pipes.  Wrought-iron  pipes  not  coated.  Glass  and 
lead  pipes.  Pipes  lined  with  smooth,  concrete, 

v  =  82.7  d°-68.S*, 
Q  =  65.01  d*-68S&. 

m  =  0.68.     Pipes  lined  with  cement  or  smooth  concrete,  pipes 
of  planed  or  rough  staves,  cast  and  wrought-iron 
pipes,  coated  or  not  coated,  wrought-iron   and 
steel-riveted  pipes  not  exceeding  36"  in  diameter 
v  =  74.1d°'695S&, 
Q  =  58.2d2'695^. 
(All  some  time  in  use  but  fairly  clean.) 

m  =  0.57.  Sewer  pipe.  Conduits  lined  with  common  brick- 
work. 

v  =  52.5  d°'70  S&, 
Q  =  41.27  d2'7  S&. 

m  =  0.53.     New  asphalt-coated  steel-riveted  pipe  exceeding  36" 
in  diameter. 
v  =  50.77  d°'7  S&, 
Q  =  39.875  d2'7  S&. 

M=  0.45.     Old  cast  and  wrought-iron  pipes  of  all  descriptions, 
not  very  clean.     Pipes  of  riveted  steel  exceeding 
36"  in  diameter,  in  use  for  several  years. 
v  =38.16  d0-87^, 
Q  =29.96  d2-87^. 

M=  0.30.    Old  pipes  of  riveted  steel  exceeding  36"  in  diameter. 
v  =  33.23d°'68^, 
=  26. 10  d2'68  S*. 


126  THE   FLOW   OF   WATER 

It  will  be  observed  that  the  powers  of  d  to  which  Velocities 
and  Quantities  are  proportional  vary  with  (a)  the  coefficient 
of  variation  of  c. .  The  difference  between  the  values  of  the 
powers  of  d  is  equal  to  0.04  between  the  successive  values  of  (a). 
For  m  =  0.68  we  have  for  instance: 
For  a  =  v*  d0'™,  d2'™, 

a  =  v^     d°'655,     d2'655, 

a  =  1.0    d0'615,     d2'615, 

a  =  -V     d0'575,    d2'575. 

v™ 

Sewers. 
B. 

For  sewers  of  circular  section  the  general  equations  for  velocity 
and  quantity  are: 

for  a  -  W», 

v  =  71.41  -  44.0     (1  -  m)  d*  S&, 
Q  =  56.08  -  34.55  (1  -  m)  &  S&. 
and  for  the  egg-shaped  section 

v  -  78.68  -  47.7    (1  -  m)  d'  S&, 
Q  =  92.76  -  56.24  (1  -  m)  d"  S&, 
d  being  the  horizontal  diameter. 

The  practically  useful  coefficients  of  roughness  for  sewers  are 
as  follows: 
m  =  0.83,  x  =  0.68,  y  =  2.68,   smooth   concrete,   very  good 

brickwork,  brickwork  washed  with  cement. 
m  =   0 . 70,  x  =  0 . 69,  y  =  2 . 69,  good  concrete,  fairly  good  brick- 
work, very  well  laid  sewer  pipe. 
m  =  0 . 57    common  concrete  or  brickwork,  common  sewer  pipe. 

Sewers  of  all  descriptions  become  in  the  course  of  a  few  years, 
frequently  in  the  course  of  a  few  months,  coated  with  sewer 
slime.  Sewage,  moreover,  does  not,  on  account  of  its  greater 
viscosity  or  stickiness,  flow  with  the  same  velocity  as  pure  water. 
The  most  reliable  data  pertaining  to  flow  in  sewers  of  all  descrip- 
tions some  time  in  use  indicate,  that  a  value  of  m  greater  than 
0 . 57  cannot  be  safely  taken. 


SEWERS  127 

For  m  —  0 . 57  we  have  in  particular 

a  =  v&, 

v  =  52.55  d°'7 

Q  =  41.27  d2'7 
for  the  circular  and 

v  =  57.43  d9'7 

Q  =  67.70  d*'7  S&, 
for  the  egg-shaped  section. 

For  a  =  1 . 0  we  have  for  the  same  value  of  m, 

v  =  42.19  d0'88^, 
Q  =  33.14  d2'88^, 
for  the  circular  and 

v  =  46.30  d0'66^, 

Q  =  53.11  d2'66  £*  for  the  egg-shaped  section. 
In  all  these  equations  the  horizontal  diameter  is  assumed  to 
be  §  of  the  vertical  diameter  or  the  vertical  diameter  H  the 
horizontal  diameter.  For  long  sewers  with  easy  curves  the 
equations  corresponding  to  a  =  V1*  should  be  taken,  for  sewers 
with  many  sharp  curves,  sharp  angles,  etc.,  for  sewers  discharg- 
ing under  water  against  a  hydraulic  counterpressure  the  equa- 
tions given  under  a  =  1 . 0  give  the  best  results. 

Comparing  the  constants  given  for  circular  and  egg-shaped 
sewers  we  find  for  equal  horizontal  diameters  and  a  =  1.0  the 
relation 

Velocity  egg-shaped  section  _  46.30 
Velocity  circular  section        42.19 
Discharge  egg-shaped  section  _  53.11  _  .   ^ 

Discharge  circular  section        33.14 

The  velocity  for  the  egg-shaped  section  is  consequently  1 . 097 

times  and  the  discharge  1 . 602  times  that  of  the  circular  section. 

It  would,  however,  be  very  erroneous  to  conclude  that  for  an 

equal  velocity  an  egg-shaped  sewer  should  have  a  horizontal 

diameter  of  =0.911  and  for  an  equal  discharge  a  horizon- 

J-  •  \)  t7  / 

tal  diameter  of  —     -  =  0 . 622  times  the  diameter  of  the  circular 
1 .502 

sewer. 


128  THE   FLOW   OF  WATER 

For  equal  velocities  the  relation  between  the  horizontal 
diameter  of  an  egg-shaped  section  and  the  diameter  of  a  circular 
section  is  given  by: 

Horizontal  diameter  egg-shaped  section 

'42 . 19  d0'66  circular  sectionY'515 
46.30  / 


and  for  equal  discharges  by: 
Horizontal  diameter  egg-shaped  section 

'33.14  cP'66  circular  sectionY'378 
53.11  / 


Assuming  a  circular  sewer  to  have  a  diameter  of  3  feet,  an 
egg-shaped  sewer  will  have,  for  the  same  velocity  a  horizontal 
diameter  of 


or  0.889  the  diameter  of  the  circular  sewer. 

For  an  equal  discharge  the  diameter  of  the  egg-shaped  sewer 
will  be 

/33.14  X  18.59V'376      0  ,10  , 
(         53.11        )       =2.  512  feet, 

or  0.834  times  the  diameter  of  the  circular  section. 

For  a  diameter  of  12  feet  we  compute  the  relations,  for  equal 
velocities 

d  =  10.42  egg-shaped  section         0_A1 
~12~  °-8701' 

for  equal  quantities 

d  ==  10  .  10  egg-shaped  section  _  -  _ 

\-jLl 

Consequently  we  may  say,  that  an  egg-shaped  sewer  should 
have  for  an  equal  velocity  a  diameter  of  0  .  88  =  |  times  the 
diameter  of  the  circular  section  and  for  an  equal  discharge  a 


OPEN   CONDUITS 


129 


diameter  of  0.834  =  |  times   the   diameter   of   the   circular 
section. 

C.    Open  Conduits. 

I.   FORM  OF  THE  CROSS-SECTION  MOST  FAVORABLE  TO  FLOW. 

The  form  of  cross-section  most  favorable  to  flow  is  one  whose 
top  width  is  equal  to  the  two  side  slopes,  whose  top  and 
bottom  width  together  are  equal  to  the  wet  perimeter,  and 
whose  mean  hydraulic  radius  is  equal  to  one  half  the  depth. 
The  semisquare  is  the  simplest  form  fulfilling  these  condi- 
tions, and  the  areas  of  the  trapezoids  may  be  expressed  in 
terms  of  the  areas  of  this  standard.  Areas  of  trapezoids  and 
the  semi-circle  thus  expressed  are  found  in  the  following 
table : 


Form  of  Section. 

Ratio  of 
Side  slopes 

Propor- 
tional 
Areas. 

Bottom 
width  in 
terms  of 
depth. 

Semisquare 

0:1 

1  .0 

2d 

Trapezoid                                 .... 

J: 

0.906 

1  .562d 

(i 

4  ' 

0.868 

1  .236d 

it 

|; 

0.875 

1  .Qd 

ti 

1: 

0.914 

0  .828d 

it 

H: 

0.976 

0  .702d 

l|: 

.053 

0  .606d 

If: 

.141 

0  .532d 

2: 

.236 

0  A72d 

2*: 

.337 

0  .424d 

2*: 

.4425 

0  .385d 

2|: 

1  .525 

0  .354d 

3: 

1  .6625 

0  .325d 

Semicircle               .        

0  .7854 

FIG.  7. 


130  THE  FLOW  OF   WATER 

The  area  of  a  trapezoid  is  equal  to 

A  =  d(b  +  d  tang  a); 
the  wet  perimeter  to 

P  =  b  +  2  d  Vl  +  tang2  a; 
the  mean  hydraulic  radius  to 

^  =       d(b  +  c?tang  q) 

b  +  2d  Vl  +  tang2  a' 
the  bottom  width  to 

7       Area 

&  =  — -7-   -  d  tang  a; 

the  top  width  to 

T      Area    .    ,  , 

•*       —~   +  a  tang  a. 


OPEN    CONDUITS 


131 


General  Relations  between  the  Velocity,  the  Discharge  and 

the  Depth  of  Water  in  the  Form  of  Section 

most  Favorable  to  Flow. 

Computing  the  velocities  of  flow  for  two  conduits  having 
mean  hydraulic  radii  equal  to  0.1  foot  and  10.0  feet,  respectively, 
from  the  general  formula 

v  =  [66  (Vr  +  m)\/r~7^]*, 

and  using  the  values  of  v  thus  found  in  the  equation 

.    log  v,  -  log  v0 
"  log  R,  -  log  RQ 

We  find  for  the  powers  of  R,  to  which  the  velocity  is  propor- 
tional, the  values  tabulated  below: 


Power  of  R  or  c?. 

a  ues£  m 

j 

1 

1 

a-F 

a=FTs 

a  =  1  .0 

a=yS 

a  =  —  - 
F» 

m  =  1  .0    —  0  .95 

0.69 

0.67 

m  =  0  .83  -  0  .80 

0.70 

0.68 

m  =  0  70  —  0  .68 

071 

0.69 

m  =  0  .57 

0.70 

m  =  0  .45 

0.715 

m  =  0  .30 

0.735 

m  =  0.15 

0.75 

m  =  0  .0  £  =  1  .0 

075 

071 

K  =  1  .25 

0765 

0.725 

K  =  1  .50 

0.775 

0735 

070 

K  =  2.0 

0.795 

0755 

071 

Computing  the  velocities  of  flow  for  semisquares  one  foot  in 
depth  from  the  general  equation  we  find  the  following  exponen- 
tial equations: 
For  a  =  y^, 

v  =  129       -  73  (1  -  m)d*S&; 


for 
for 

for 


v  =  111.6  -  63.3  (1  - 

a  =  1.0, 
v  =  85.86  -  46.61  (1  -  m)  d*  S*\ 

a  = 


v  =  67.62  -  35.4  (1  -  m)d* S**. 


132  THE    FLOW   OF   WATER 

2 

In  any  of  these  equations  the  term  2  — may  be  sub- 

JL  ~T~  K> 

stituted  for  its  equivalent  1  —  m.     In  particular  we  have  for 
the  semicircle: 

m  =  1.0    Semicircular  channels  lined  with  clean  cement, 
v  =  129  d°'69  S&. 

m  =  0.83  Semicircular   channels    of    brickwork   washed    with 
cement, 

v  -  116  d°'7  S&. 

m  =  0.70  Semicircular  channels  lined  with  rough  boards, 

v  =  104. 5  d°'71  S&. 
For  the  semisquare  we  have: 

a  =  7*. 
m  =  0.95  Channels  lined  with  clean  cement,  planed  boards, 

v  =  108.43  d°'nS&, 
Q  =  216.86d2'67£T7. 

m  =  0.80  Channels   lined   with   smooth   concrete,   very   good 
brickwork, 

v  =  98.87  d°'MS^t 
Q  =  197.74  d2'68  S&. 

m  =  0 . 70  Channels  lined  with  sawed  boards  or  good  brickwork, 
v  =    92.48d°'69/ST97, 
Q  =  184.96  d™°  S&. 

m  =  0.57  Channels  lined  with  common  brickwork,  rough  con- 
crete or  very  good  ashlar, 

v  =    84.23  d°'n 

Q  =  168.46  d2'70 
m  =  0.45  Channels    lined    with    rough    brickwork,    common 
ashlar  or  very  rough  concrete, 

*-    76.67  d°'™ 

Q  =  153.34  d2'715 
m  —  0.30  Channels  lined  with  good  rubble  masonry, 

v  =    67.27  d*-mS& 

Q  =  134.54  d*' 


OPEN  CONDUITS  133 

m  =  0.15  Channels  lined  with  roughly  hammered  masonry, 
channels  in  cemented  gravel  up  to  one  inch  in 
diameter, 

v  =  57.8  d°'75  S&, 
Q  =  114.6  d2'75  S&, 
a  =  1.0. 

m  =  0 . 0  Channels  lined  with  common  rubble  masonry,  tunnels 
K  =  1.0  in  rockwork,  channels  in  cemented  gravel  exceeding 
one  inch  in  diameter, 

v  =  39.24  d°'75  £4, 
Q  =  78.48  d2'75  S*. 

m  =  —  0.1  Fairly  regular  channels  in  loose  sand,  or  sand  with 
K  =      1.2  gravel  imbedded 

v  =  35.39  d*'mS*, 

Q  =  70.78  d2-785^. 

m  =  -  0.2  Fairly  regular  channels  in  earth,  free  from  debris  or 
K  =      1.5  vegetation, 

v.  =  30.86  d°'775  S*, 

Q  =  61.72  d2-775£i 

m  =  —  0.32    Channels  in  earth  with  debris  or  vegetation, 
K=      1.93 

v  =  26.1  d°'™  S*, 

Q  =  52.2  d2'795Si 

If  the  cross-section  of  the  conduit  is  a  trapezoid,  the  dis- 
charges are  multiplied  by  the  proportional  areas  found  in  the 
table  given  at  the  beginning  of  this  chapter. 

For  a  trapezoid  having  side  slopes  of  1: 1,  for  instance,  the 
discharges  found  from  any  of  the  equations  given  above  are  mul- 
tiplied by  0 . 914,  for  side  slopes  of  2  :  1  by  1 . 236  etc.  The  depth 
being  the  same,  the  velocity  is  not  affected  by  the  side  slope. 

Values  of  the  powers  of  d  relating  to  Velocities  are  found  in 
Table  E ;  those  relating  to  Quantities  of  discharge  in  Table  F. 

Values  of  the  sines  of  the  slope  and  their  roots  are  found  in 
Table  C. 


134  THE  FLOW    OF   WATER 

For  channels  in  earth,  in  case  the  velocity  exceeds  the  limit 
where  erosion  begins,  the  following  equations  may  be  used: 

m  =  -  0.1  v  =  28.68  d°'™ 

K  =       1.20  Q  =  57.36  d2'™ 

m  =  -  0.20  v  =  25.14  d°'735 

K=       1.50  0  =  50.28  d2'™ 

m  =  -  0.32  v  =  20.90  d°'755 

K=       1.93  Q-41.80  d2'm 

It  is,  however,  more  convenient  to  use  the  equations  previ- 
ously given,  for  which  the  powers  of  d  and  S  are  found  in  the 
tables,  and  multiply  the  Velocities  or  Quantities  found  from  the 
formulas  by  the  coefficient  of  variation  of  C,  which  in  these 

cases  is  equal  to  a  =  — -  -     Values  of  a  =  — -  are  found  in 
yh 

column  10,  Table  V. 


II.   GENERAL   EQUATIONS. 

In  the  design  of  cross-sections  of  channels  it  is  not  always 
possible  to  use  the  form  of  section  most  favorable  to  flow.  Other 
forms  are  frequently  required  for  special  purposes,  are  constructed 
at  less  cost,  or  offer  other  advantages. 

For  wooden  flumes,  for  instance,  the  triangular  section  is  fre- 
quently adopted. 

If  the  sideslopes  of  a  triangle  are  1  : 1,  or  the  sides  inclined  45° 
(which  is  the  usual  sideslope  for  a  flume),  the  area  of  its  cross- 
section,  its  mean  hydraulic  radius  and  consequently  its  velocity 
and  discharge  are  equal  to  those  of  a  semisquare  when 

(1)  the  depth  =  Varea  of  semisquare. 

(2)  the  top  width  =  \/4  X  area  of  semisquare. 

In  the  design  of  channels  in  earth  it  is  frequently  necessary, 
in  order  to  keep  the  velocity  below  the  eroding  limit,  to  make 
the  sections  wide  and  shallow,  so  that  the  frictional  resistance 
may  be  increased  and  the  flow  retarded.  In  many  cases  a 
shallow  section  is  also  more  easily  constructed  and  at  less  cost. 


OPEN   CONDUITS  135 

The   general   exponential  equations,    derived  as  previously 
indicated,  are  as  follows: 

a  =  7*, 

v  =  243      -  131  .6  (1  -  m)  r*  SA, 
a 


v  =  205.8  -  112      (1  -  m)  r*  S 

a  =  7* 

v  =  176.3  -  93.5    (1  -  m)  r*<S^, 
a  =  1.0, 

v  =  132       -  66        (1  -  m)  r*  5*, 
a  = 


v  =  100.6  -  48.3   (1  -  m)  r* 

o 
In  any  of  these  equations  2  -  -  -  ^may  be  substituted  for 

(1  -  m). 
The  practically  most  useful  special  equations  are  as  follows: 


a 

=  71?* 

m  =  0.83 

v  =  186.7^ 

a 

=  yrs 

m  =  0.70 

v  =  172.3 

a 

=  yi* 

m  =  0.95 

v  =  171.6 

a 

—  yfa 

m  =  0.80 

v  =  158.2 

a 

=  yA 

m  =  0.70 

v  =  148.1 

a 

=  yiV 

m  =  0.57 

v  =  136.1 

a 

=  y& 

m  =  0.45 

v  =  121.9 

a 

=  yi* 

m  =  0.30 

v  =  110.8 

a 

=  yiV 

m  =  0.15 

v  =    96.8 

a 

=  1.0 

m  =  0.0 

v  =    66.0 

a 

=  1.0 

K  =  1.2 

v  =    60.0 

a 

=  1.0 

K  =  1.5 

v  =    52.8 

a 

=  1.0 

K  =  1.93 

v  =    45.0 

r°'71 


S* 


It  will  be  observed  that  the  constants  of  these  equations  are 
equal  to  those  given  for  the  semisquare  and  the  semicircle, 
multiplied  by  2*  =  2°-67,  2°'68,  2°'69,  etc.,  as  the  case  may  be. 


136  THE   FLOW   OF   WATER 

Explanation  of  the  Use  of  the  Tables  of  Velocities  and 
Quantities  Q,  H,  and  I. 

Table  G  contains  the  quantities  of  discharge  in  cubic  feet  per 
second  of  a  conduit  one  foot  in  diameter,  for  seven  different 
degrees  of  roughness  and  174  slopes. 

For  the  discharge  in  gallons  per  second  multiply  the  quanti- 
ties found  in  the  table  by  7.48052.  For  the  velocity  of  flow 

1  14 

multiply  the  quantities  found  in  the  table  by  =  —  or, 

0 . 78o4       1 1 

if  the  conduit  is  egg-shaped  by          •  =  f  nearly. 

I. 

To  find  the  quantity  of  discharge  for  any  diameter,  and  a 
given  slope,  multiply  the  value  of  Q  found  on  line  with  the  given 
slope  under  the  value  of  m}  which  indicates  the  particular  degree 
of  roughness  of  the  conduit,  by  the  value  of  dy  found  in  Table  D 
under  the  same  value  of  m. 

Example:  What  is  the  discharge  of  a  cast-iron  conduit  same 
time  in  use;  the  diameter  being  36  inches,  and  the  slope  1  : 10,000? 

In  Table  G,  in  column  under  m  =  0 . 68,  in  line  with  S  =  0 . 0001 
we  find  Q  =  0.3349/3  per  second. 

In  Table  D,  under  dy  =  d2-695,  we  find  for  d  =  36",  d2-695  = 
19.31.  Hence  Q  =  0.3349  X  19.31  =  6.54 f  per  second. 

II. 

To  find  the  loss  of  head  or  the  slope  corresponding  to  a  given 
discharge  and  a  given  diameter,  divide  the  given  discharge  by 
the  value  of  dy  found  in  Table  D  as  indicated  above.  The  quo- 
tient then  found  will  indicate,  in  Table  G,  under  the  proper  value 
of  m,  the  slope  required  to  produce  the  given  discharge. 

Example:  What  is  the  loss  of  head  corresponding  to  a  dis- 
charge of  55  /3  per  second,  the  conduit  being  a  new,  asphalt- 
coated,  steel-riveted  pipe  6  feet  in  diameter? 

In  Table  D,  under  dy  =  d2'7,  we  find  for  a  diameter  of  72" 
d2'70  =  126.18.  Dividing  55  by  126.18  the  quotient  0.436  is 


OPEN    CONDUITS  137 

the  quantity  of  discharge  of  a  conduit  one  foot  in  diameter  for 
the  required  slope. 

In  Table  G,  under  m  =  0.53,  the  value  of  Q  coming  nearest 
to  0.436  is  0.439.  This  stands  in  line  with  S  =  0.0002,  conse- 
quently S  =  0.0002  is  the  slope  required  to  produce  the  given 
discharge. 

III. 

To  find  the  diameter  corresponding  to  a  given  discharge  and 
a  given  slope,  divide  the  given  discharge  by  the  discharge  of  a 
conduit  one  foot  in  diameter  for  the  given  slope  as  found  in 
Table  G.  The  quotient  is  the  value  of  dy  for  which  the  diameter 
is  found  in  Table  D. 

Example:  What  will  be  the  horizontal  diameter  of  an  egg- 
shaped  sewer,  the  discharge  being  200  /3  per  second,  and  the 
slope  1  : 2500? 

In  Table  G,  in  column  headed  "Egg-shaped  section,"  and  in 
line  with  S  =  0.0004,  we  find  Q  =  1.0759.  Dividing  200  by 
1.0759  the  quotient  is  185.8. 

In  Table  D,  in  column  headed  d2'70  the  nearest  value  above 
185.8  is  191.3,  which  stands  in  line  with  d  =  7.0  feet;  hence 
7  feet  is  the  horizontal  diameter  required  to  produce  the  given 
discharge. 

IV. 

To  find  the  loss  of  head  or  the  slope  required  to  produce  a 
given  velocity  for  a  given  diameter,  divide  the  given  velocity 
by  the  value  of  dx  found  in  Table  D,  as  indicated  above.  The 
quotient  thus  found,  multiplied  by  0.7854  =  -^  will  indicate, 
in  Table  G,  under  the  proper  value  of  m,  the  slope  required. 

Example:  What  is  the  proper  slope  for  an  8-inch  sewer  pipe? 

For  well  constructed  sewers  the  permissible  velocity  is 

0.25 


v  =  2  + 


d 


which  gives  for  an  8-inch  sewer  v  =  2.375  feet  per  second. 

In  Table  D,  in  column  headed  dx  =  d0'7,  in  line  with  d  =  8 
inches,  we  find  d*  =  0.7529.  Dividing  2.375  by  0.7529  the 
quotient  is  3.053,  which  multiplied  by  ^  gives  Q  =  2.477. 


138  THE   FLOW   OF   WATER 

In  Table  G,  in  column  headed  m  =  57,  we  find  the  nearest 
value  of  Q  above  2.477  to  be  2.495,  which  is  in  line  with 
S  =  0 . 005,  which  is  the  slope  required. 

V. 

In  case  the  conduit  is  egg-shaped,  proceed  as  before,  but 
instead  of  multiplying  by  0 . 7854  =  ^£  multiply  by  1 . 147  =  f . 

Example:  What  is  the  least  permissible  slope  for  an  egg- 
shaped  sewer  having  a  horizontal  diameter  of  10  feet? 

In  this  case:  v  =  2  +  ^P  =  2.025  feet  per  second. 

In  Table  D,  in  column  headed  dx  =  d0'7  and  in  line  with 
d  =  120  inches,  we  find  d0'7  =  5.011.  Dividing  2.025  by 
5.011  the  quotient  is  0.404,  which  multiplied  by  1.147  gives 
Q  =  0.473. 

In  Table  G,  in  column  headed  "Egg-shaped  section,"  we  find 
the  nearest  value  of  Q  above  0.473  to  be  0.4738,  which  is  in 
line  with  S  =  0.000085,  hence  S  =  0.000085  is  the  least  per- 
missible slope  for  a  10-foot  egg-shaped  sewer. 

VI. 

In  Table  H  we  find  velocities  of  flow  in  a  semisquare  one 
foot  in  depth  for  the  practically  most  useful  values  of  m  or  K 
and  174  slopes. 

Table  H  applies  to  any  trapezoid  having  the  form  of  section 
most  favorable  to  flow. 

To  find  the  velocity  of  flow  corresponding  to  any  depth  what- 
soever, either  in  the  semisquare,  or  the  trapezoid  having  the 
form  of  section  most  favorable  to  flow,  multiply  the  values 
found  in  the  table  by  the  values  of  dx  found  in  Table  E. 

Example:  What  is  the  velocity  of  flow  in  a  channel  in  earth 
having  the  form  of  section  most  favorable  to  flow,  the  bed  of 
the  channel  being  covered  with  stones,  the  depth  10  feet  and  the 
slope  1  : 10,000? 

In  Table  H  under  K  =  1.93,  and  in  line  with  S  =  0.0001, 
we  find  v  =  0.261. 

In  Table  E,  in  column  headed  K  =  2.0,  and  in  line  with 
d  =  10,  we  find  d°'795  =  6.238.  Multiplying  the  two  quantities, 
we  find  v  =  1.628  feet  per  second. 


OPEN   CONDUITS  139 

VII. 

Remembering  that  in  the  form  of  section  most  favorable  to 
flow  2R  =  d;  or  the  mean  hydraulic  radius  equal  to  one  half 
the  depth,  it  is  plain  that  Table  H  can  also  be  used  to  find  the 
velocity  of  flow  in  channels  not  having  the  form  of  section  most 
favorable  to  flow.  It  is  only  necessary  always  to  consider 
R  =  %  d,  and  multiply  or  divide  by  the  value  of  d,  which  corre- 
sponds to  the  given  value  of  R. 

Example:  What  is  the  velocity  of  flow  in  a  channel  lined 
with  common  brickwork,  the  slope  being  1  : 10,000  and  the 
mean  hydraulic  radius  6  feet? 

In  Table  H,  in  column  headed  m  =  0.57,  and  in  line  with 
S  =  0.0001,  we  find  v  =  0.6424. 

R  =  6  is  equal  to  d  =  12. 

In  Table  E,  in  column  headed  m  =  0.57,  and  in  line  with 
d  =  12,  we  find  D0'7  =  5.695.  Multiplying  the  two  quantities 
we  have  v  =  3.658  feet  per  second. 

Example:  What  is  the  slope  required  for  a  velocity  of  8  feet 
per  second,  the  conduit  being  a  triangular  flume  of  sawed  boards 
and  the  mean  hydraulic  radius  equal  to  2  feet? 

R  =  2  is  equal  to  d  =  4 . 0. 

In  Table  E,  in  column  headed  m  =  0.70  and  in  line  with 
d  =  4.0,  we  find  d0-69  =  2.603.  Dividing  8  by  2.603,  the 
quotient  is  3.073,  which  is  the  value  of  v  corresponding  to  a 
depth  of  one  foot. 

In  Table  H,  in  column  headed  m  =  0.70,  we  find  the  value 
nearest  to  3.073  to  be  3.061,  which  is  in  line  with  S  =  0.0016, 
which  is  the  required  slope. 

Example:  What  is  the  value  of  the  mean  hydraulic  radius 
required  to  produce  a  velocity  of  2.8  feet  per  second,  the  slope 
being  1  : 10,000  and  the  conduit  a  channel  in  earth  in  good  con- 
dition, free  from  stones  and  plants? 

In  Table  H,  in  column  headed  K  =  1.5  and  in  line  with 
S  =  0.0001  we  find  v  =  0.3086.  Dividing  2.8  by  0.3086  the 
quotient  is  9.073. 


140  THE   FLOW   OF   WATER 

In  Table  E,  in  column  headed  K  =  1.5,  we  find  the  nearest 
value  above  9 . 073,  to  be  9 . 191,  which  stand  in  line  with  d  =  17 . 5. 

17  5 
Hence  R,  the  mean  hydraulic  radius  required,  is  — ^-  =8 . 75  feet. 

2 

VIII. 

As  Table  H  holds  good  for  all  conduits  having  the  form  of 
section  most  favorable  to  flow,  it  is  evident  that  it  may  be 
used  to  find  velocities  of  flow  in  circular  conduits  running  full. 

To  find  velocities  corresponding  to  any  diameter,  it  is  necessary 
to  keep  in  mind  the  fact,  that  the  depth  in  a  semicircle  is  one 
half  the  diameter,  and  multiply  or  divide  by  the  value  of  d 
which  corresponds  to  the  semidiameter. 

Example:  What  is  the  least  permissible  slope  for  a  6-inch 
sewer  pipe? 

Here  the  semidiameter  or  the  depth  is  3  inches.     The  per- 

0  25 

missible  velocity  is  v  =  2  +        -  =  2.5  feet  per  second. 

a 

In  Table  D,  in  column  headed  dx  =  d0'7,  we  find  in  line  with 
d  =  3  inches,  d0'7  =  0.3703.  Dividing  2.5  by  0.3703,  the 
quotient  is  6.751. 

In  Table  H,  under  m  =  0.57,  the  value  of  v  coming  nearest 
to  6.751  is  6.757,  which  is  in  line  with  s  =  0.0085.  Hence 
s  =  0 . 0085  is  the  least  permissible  slope  for  a  6-inch  sewer  pipe. 

IX. 

For  the  classes  of  circular  conduits  whose  degrees  of  roughness 
are  indicated  by  the  coefficient  m  =  0.95,  m  =  0.83,  m  =  0.68. 
Table  H  gives  velocities  also  in  case  the  conduit  is  between  300 
and  1000  diameters  in  length,  or  has  sharp  elbows,  such  as  city 
mains. 

Example:  What  will  be  the  velocity  of  flow  in  a  city  main 
3  feet  in  diameter,  the  slope  being  1  : 200? 

Here  the  semidiameter  is  1 . 5  feet. 

The  difference  in  the  powers  of  d  to  which  the  velocity  is  pro- 
portional between  a  =  V^  and  a  =  V™  is  equal  to  0 . 04.  For 
m  =  0.68  or  0.70  and  a  =  V*,  we  have  for  the  power  of  d, 
x  =  0.705,  consequently  for  a  =  V&  x  =  0.665. 


OPEN   CONDUITS  141 

In  Table  D,  in  line  with  d  =  1.5,  we  find  d°'66  =  1.307, 
d0'67  =  1.312,  hence  d°'665  -  1.3095. 

In  Table  H,  in  column  headed  m  =  0 . 70  (which  is  sufficiently 
equal  to  0.68  to  apply  in  such  cases)  and  in  line  with  S  =  0.005, 
we  find  v  =  5.595.  Multiplying  we  have  1.3095x5.595 
=  7.3266  feet  per  second.  The  discharge  will  be  Q  =  7.3266 
X  9  X  0.7854  =  51.79  cubic  feet  per  second. 

X. 

Table  I  gives  the  Quantities  of  discharge  in  cubic  feet  per 
second  of  a  semisquare  one  foot  deep  for  the  practically  most 
useful  values  of  m  or  K  and  174  slopes. 

For  the  trapezoids  or  the  semicircle  the  quantities  given  in 
the  table  are  to  be  multiplied  by  their  proportional  areas. 

Example:  What  is  the  discharge  of  a  channel  lined  with  dry 
rubble  masonry,  or  a  channel  in  rockwork,  or  a  channel  in  coarse 
cemented  gravel,  the  side  slopes  being  one  half  to  1,  the  depth 
12  feet,  and  the  sine  of  the  slope  0.0005? 

In  Table  I7  in  column  headed  m  =  0.0,  and  in  line  with 
S  =  0.0005  S,  we  find  Q  =  1.755. 

In  Table  F,  in  column  headed  m  =  0.0,  and  in  line  with 
d  -  12,  we  find  d2'75  -  928.4. 

For  a  sideslope  of  i  :  1  the  proportional  area  is  0 . 868.  Mul- 
tiplying the  three  quantities  1.755  X  928.4  X  0.868,  we  find 
Q  =  1414.2  /3  per  second. 

Example:  What  will  be  the  dimensions  of  a  channel  in  sand, 
the  discharge  being  200  cubic  feet  per  second,  the  slope  1 :  10,000 
and  the  sideslopes  3:1?  For  a  sideslope  of  3  : 1  the  propor- 
tional area  is  1.6625.  Dividing  200  by  1.6625  we  have  for 
the  discharge  of  a  semisquare  of  equal  depth  Q  =  120 . 3  feet. 

In  Table  I,  in  column  headed  K  =  1.2  and  in  line  with 
S  =  0.0001,  we  find  Q  =  0.708.  Dividing  120.3  by  0.708 
the  quotient  is  169.9. 

In  Table  F,  in  column  headed  K  =  1.25,  we  find  the  value 
nearest  to  169.9  to  be  169.5,  which  stands  in  line  with  6.4  feet, 
which  is  the  depth  required. 

The  bottom  width  of  the  conduit  will  be  6.4  X  0.325  =  2.08 
feet,  the  top  width  6. 4x3x2  +  2. 08  =  40. 48  feet. 


142  THE  FLOW   OF   WATER 

The  cross-section  will  be  6.4  X  -  -  =136. 2  square 

200 

feet,  and  the  velocity  =1.46  feet  per  second. 

lob .  2i 

XI. 

In  the  design  of  channels  in  earth,  especially  those  in  light 
soils,  it  is  necessary  to  keep  the  velocities  of  flow  within  the 
eroding  limits.  For  channels  in  light  sandy  soils  a  velocity 
exceeding  1.5  feet  per  second  should  not  be  allowed,  for  chan- 
nels in  earth  with  some  clay  2 . 5  feet  per  second  should  be  the 
limit. 

To  keep  the  velocities  down  two  methods  may  be  used : 

(1)  The  slope  may  be  reduced  by  means  of  weirs,  dams  and 
drops. 

(2)  The  mean  hydraulic  radius  may  be  reduced  by  making 
the  channel  wide  and  shallow. 

Example:  A  channel  in  sandy  soil  is  to  carry  500  cubic  feet 
per  second,  the  velocity  is  not  to  exceed  1.5  feet  per  second, 
the  sideslopes  are  to  be  3 : 1  and  the  depth  of  the  water  8  feet. 
What  will  be  the  slope  of  the  channel? 

The  area  of  the  cross-section  will  be  —  =  333.3  feet. 

1 . 5 
qqq   q 

The  mean  width  -  ^  =  41.66  feet. 

o 

The  bottom  width  41.66-8  X  3  =  17.66  feet. 

The  wet  perimeter  17.66  +  2  V82  +  (8  X  3)2  =  68.26  feet. 

333  3 

The  mean  hydraulic  radius  — —  -  =  4.883  feet. 

bo.zb 

R  =4.883  corresponds  to  d  =  9.766. 

In  Table  E,  in  column  headed  K  =  1 .25,  we  find  the  value  of 
d°-765for9.8tobe5.732and5.687for9.7,meanfor9.75  =  5.71. 
Dividing  1.5  by  5.71  the  quotient  is  0.2627. 

In  Table  H,  in  column  headed  K  =  1 . 20,  we  find  the  value 
coming  nearest  to  0.2627  to  be  0.2625,  which  stands  in  line 
with  S  =  0 . 000055.  This  is  equal  to  a  fall  of  0 . 29  feet  per  mile. 

Example:  A  channel  in  sandy  soil  is  to  carry  500  cubic  feet 
per  second  at  a  velocity  of  1.5  feet  per  second.  The  sideslopes 
are  to  be  3  : 1  and  the  slope  of  the  channel  0 . 5  feet  per  mile,  or 


OPEN    CONDUITS 


143 


0  .  000095.  What  will  be  the  depth  of  the  channel  and  its  bottom 
width? 

In  Table  H,  in  column  headed  K  =  1.2  and  in  line  with 
S  =  0.000095,  we  find  v  =  0.345.  Dividing  1.5  by  0.345  the 
quotient  is  4.347. 

In  Table  E,  under  K  =  1.25,  we  find  the  value  of  d°'765  next 
above  4.347  to  be  4.383,  which  stands  in  line  with  d  =  6.9. 
Hence  R  =  3.45. 

If  the  channel  is  given  a  depth  of  4  feet,  its  mean  width  will  be 

3-5^  =  83.33  feet; 
4 

its  bottom  width   83.33  -  4  X  3  =  71.33  feet; 

its  wet  perimeter  71.33  +  2  \/42  +  (4  X  3)2  =  96.53  feet; 

333  3 

its  mean  hydraulic  radius  '    =  3.45  as  above. 


TABLE  C. 

SINES  OF  SLOPES  AND  ROOTS  OF  SINES  OF  SCOPES. 


S 

s& 

Sfs 

B* 

si 

S 

S& 

ST6T 

Sfr 

8* 

000025 

002580 

00309 

00366 

005 

00050 

01390 

01583 

01779 

02236 

000030 

00284 

00341 

00402 

00548 

00055 

01467 

01668 

01882 

02346 

000035 

00311 

00371 

00437 

00591 

00060 

01541 

01748 

01970 

02450 

000040 

00336 

00399 

00469 

00631 

00065 

01612 

01826 

02054 

02550 

000045 

00359 

00425 

0050 

00671 

00070 

01680 

01902 

02136 

02645 

000050 

00381 

00451 

00528 

00707 

00075 

01747 

01975 

02216 

02739 

000055 

00402 

00475 

00556 

00741 

00080 

01806 

02045 

02287 

02830 

000060 

00422 

00498 

00582 

00775 

00085 

01874 

02114 

02369 

02916 

000065 

00441 

00520 

00607 

00806 

00090 

01935 

02181 

02441 

030 

000070 

00461 

00541 

00631 

00836 

00095 

01995 

02247 

02511 

03082 

000075 

00478 

00562 

00655 

00866 

001 

02047 

0231 

02581 

03163 

000080 

00496 

00583 

00677 

00895 

0011 

02166 

02424 

02714 

03316 

000085 

00513 

00602 

0070 

00922 

0012 

02276 

02552 

02842 

03464 

000090 

00530 

00621 

00721 

00948 

0013 

02380 

02665 

02965 

03605 

000095 

00544 

00640 

00742 

00975 

0014 

02481 

02775 

03084 

03742 

0001 

00562 

00658 

00762 

010 

0015 

02579 

02878 

03199 

03873 

000125 

00637 

00743 

008572 

01118 

0016 

02647 

02985 

03310 

040 

00015 

00706 

00821 

00945 

01225 

0017 

02768 

03077 

03428 

04121 

000175 

00770 

00893 

01003 

01323 

0018 

02858 

03183 

03523 

04243 

00020 

00831 

00960 

01123 

01414 

0019 

02946 

03278 

03624 

04359 

000225 

00887 

01024 

01172 

015 

0020 

03033 

03374 

03725 

04472 

00025 

00941 

01085 

01248 

01581 

0021 

03117 

03462 

03822 

04583 

000275 

00993 

01140 

01303 

01658 

0022 

0320 

03552 

03918 

04691 

00030 

01043 

0120 

01365 

01732 

0023 

03280 

03639 

04011 

04776 

00035 

01138 

01301 

01480 

01871 

0024 

0336 

03724 

04103 

04898 

00040 

01226 

01442 

01589 

020 

0025 

03438 

03808 

04191 

050 

00045 

01311 

01474 

01693 

02122 

0026 

03515 

03820 

04280 

05099 

144 


THE    FLOW  OF   WATER 


TABLE   C  —  Continued. 


s 

Sh 

Si\ 

S*T 

S$ 

S 

Sft 

Siei 

Si9r 

Si 

0027 

03590 

03941 

04367 

05196 

0080 

06614 

07182 

07760 

08944 

0028 

03665 

04051 

04452 

05292 

00825 

06731 

07303 

07888 

09083 

0029 

03737 

04165 

04535 

05384 

0085 

06843 

07423 

08013 

09193 

0030 

03810 

04206 

04617 

05478 

00875 

06956 

07542 

08139 

09357 

0031 

0388 

04282 

04698 

05567 

0090 

07067 

07659 

0826 

09413 

0032 

03951 

04357 

04779 

05656 

00925 

07177 

07774 

0838 

09618 

0033 

04019 

04430 

04856 

05744 

0095 

07286 

07888 

08499 

09747 

0034 

04087 

04503 

04934 

05831 

00975 

07393 

08001 

08611 

09805 

0035 

04155 

04595 

05010 

05916 

01 

07499 

08111 

08733 

10 

0036 

04227 

04646 

05092 

060 

01025 

07604 

08222 

08848 

10117 

0037 

04287 

04716 

05159 

06083 

0105 

07708 

08330 

08962 

10247 

0038 

04352 

04785 

05233 

06164 

01075 

07818 

08440 

09032 

10369 

0039 

04415 

04853 

05305 

06245 

0110 

07912 

08544 

09185 

10488 

0040 

04479 

04921 

05377 

06310 

01125 

08012 

08650 

09295 

10600 

0041 

04541 

04987 

05446 

06402 

01150 

08113 

08753 

09361 

10724 

0042 

04603 

05053 

05519 

06481 

01175 

08211 

08857 

09512 

10840 

0043 

04665 

05109 

05586 

06557 

0120 

08309 

08959 

09620 

10954 

0044 

04726 

05187 

05655 

06633 

01225 

08406 

09061 

09722 

11068 

0045 

04786 

05248 

05722 

06709 

01250 

08502 

09161 

09828 

1118 

0046 

04845 

05314 

05789 

06782 

01275 

08597 

09261 

09939 

11292 

0047 

04904 

05374 

05855 

06855 

0130 

08691 

09360 

10034 

11368 

0048 

04962 

05435 

05922 

06928 

01325 

08785 

09457 

10136 

11511 

0049 

05020 

05497 

05986 

070 

01350 

08878 

09554 

10236 

11619 

0050 

05078 

05567 

06051 

07071 

01375 

08970 

09650 

10337 

11726 

0051 

05135 

05618 

06115 

07141 

0140 

03060 

09745 

10436 

11832 

0052 

05191 

05678 

06178 

07212 

01425 

09152 

09840 

10535 

11937 

0053 

05247 

05738 

06241 

07280 

01450 

09242 

09934 

10632 

12042 

0054 

05303 

05795 

06301 

07347 

01475 

09342 

10026 

10729 

12145 

0055 

05357 

05854 

06364 

07415 

0150 

09420 

1012 

10825 

12248 

0056 

05412 

05912 

06425 

07483 

01525 

09508 

10211 

10920 

12349 

0057 

05466 

05970 

06486 

07548 

01550 

09595 

10302 

11014 

12450 

0058 

05520 

06026 

06545 

07616 

01575 

09682 

10392 

11108 

1255 

0059 

05574 

06083 

06605 

07681 

016 

09767 

10480 

11199 

1265 

0060 

05627 

06139 

06664 

07746 

01625 

09854 

10571 

11293 

1275 

0061 

05680 

06193 

06723 

07810 

01650 

09939 

10659 

11385 

1285 

0062 

05731 

0625 

06781 

07874 

01675 

10024 

10748 

11476 

1294 

0063 

05783 

06304 

06839 

07937 

0170 

10107 

10834 

11566 

1304 

0064 

05834 

06359 

06895 

080 

01725 

10190 

1092 

11656 

1313 

0065 

05886 

06413 

06953 

08062 

01750 

10273 

11007 

11745 

1323 

0066 

05936 

06466 

07009 

08124 

01775 

10332 

11092 

11834 

1331 

0069 

05986 

0652 

07064 

08185 

018 

10437 

11177 

11921 

1342 

0068 

06036 

06573 

07121 

08246 

01825 

10519 

11261 

12009 

1351 

0069 

06086 

06625 

07176 

08307 

01850 

10600 

11346 

12096 

1360 

0070 

06136 

06677 

07231 

08367 

01875 

1068 

11429 

12182 

1369 

0071 

06186 

06739 

07285 

08427 

0190 

1076 

11512 

12268 

1378 

0072 

06234 

06781 

07339 

08485 

01925 

1088 

11594 

12353 

1388 

0073 

06283 

06832 

07393 

08544 

01950 

1092 

11676 

12437 

1396 

0074 

06331 

06883 

07447 

08602 

01975 

10946 

11757 

1252 

1406 

0075 

06379 

06933 

07500 

08660 

020 

11074 

11812 

1261 

1414 

0076 

06426 

06983 

07552 

08718 

0205 

11228 

120 

1277 

1432 

0077 

06478 

07033 

07605 

08775 

021 

11383 

12157 

1294 

1450 

0078 

06521 

07083 

07657 

08832 

0215 

11534 

12315 

1310 

1466 

0079 

06568 

07133 

07709 

08888 

0220 

11658 

1247 

1336 

1483 

OPEN   CONDUITS 


145 


TABLE  C  —  Concluded. 


s 

SA 

sft 

«A 

Si 

8 

Si9* 

SA 

S* 

Si 

0225 
0230 

11842 
11980 

1262 

1228 

1343 
1357 

150 
1517 

075 

080 

2330 
2415 

2434 
2521 

2538 
2626 

2739 

2828 

0235 

12118 

1293 

1373 

1533 

085 

2497 

2606 

2712 

2916 

0240 

12270 

1308 

1388 

1550 

090 

2581 

2689 

2795 

30 

0245 

12414 

1323 

1404 

1565 

095 

2644 

2750 

2859 

3082 

0250 

12550 

1337 

1419 

1581 

10 

2739 

2848 

2956 

3163 

030 

1391 

1478 

1562 

1732 

20 

4044 

4157 

4265 

4583 

035 

1517 

1606 

1695 

1871 

30 

5080 

5185 

5287 

5478 

040 

1635 

1728 

1820 

20 

40 

5973 

6066 

6157 

6310 

045 

1748 

1842 

1931 

2122 

50 

6771 

6852 

6928 

7071 

050 

1854 

1950 

2048 

2236 

60 

7503 

7567 

7631 

7746 

055 

1956 

2073 

2153 

2346 

70 

8182 

8232 

8279 

8367 

060 

2008 

2155 

2255 

2449 

80 

8820 

8854 

8899 

8944 

065 

2155 

2251 

2353 

2550 

90 

0  .7425 

0  .9442 

0  .9457 

0  .9487 

070 

2240 

2344 

2447 

2646 

1.00 

1.0 

1  .0 

1  .0 

1  .0 

TABLE  D. 

POWERS  OF  THE  DIAMETERS  OF  CIRCULAR,  SEMICIRCULAR  AND  EGG- 
SHAPED   CONDUITS   IN  FEET. 


|  Diameter! 
|  in  Inches.) 

rf0.66 

^2.66 

rfO-67 

d2.67 

rfO-68 

^2.68 

^0.695 

d2.695 

d°-7 

#* 

1 

0.0194 

0.00135 

0.0189 

0.00134 

0.01844 

0.00128 

0.0178 

0.00123 

0.0176 

0.00192 

2 

0.3064 

0.0085 

0.3010 

0.00835 

0.2956 

0.0082 

0.2878 

0.0080 

0.2852 

0.60772 

3 

0.4005 

0.0150 

0.3950 

0.0246 

0.3807 

0.0243 

0.3729 

0.02385 

0.3703 

0.0235 

4 

0.4842 

0.0538 

0.4790 

0.0532 

0.4738 

0.0526 

0.4660 

0.0518 

0.4635 

0.1511 

5 

0.5611 

0.0974 

0.5562 

0.0965 

0.5514 

0.0957 

0.5460 

0.0941 

0.5418 

0.0949 

6 

0.6329 

0.1582 

0.6285 

0.1571 

0.6241 

0.1561 

0.6177 

0.1544 

0.6155 

0.1534 

7 

0.7006 

0.2584 

0.6968 

0.2371 

0.6932 

0.2358 

0.6875 

0.2339 

0.6857 

0.2335 

8 

0.7652 

0.840 

0.7620 

0.3386 

0.7590 

0.3373 

0.7543 

0.3353 

0.7529 

0.3343 

9 

0.8271 

0.4656 

0.8247 

0.4644 

0.8223 

0.4630 

0.8187 

0.4610 

0.8175 

0.4602 

10 

0.8866 

0.6187 

0.8810 

0.6145 

0.8834 

0.6134 

0.8809 

0.6917 

0.8801 

0.6112 

11 

0.9442 

0.7931 

0.9433 

0.7926 

0.9424 

0.7919 

0.9413 

0.7908 

0.941 

0.7904 

12 

1.0 

1.0 

1.0 

1.0 

1.0 

1.0 

1.0 

1.0 

1.0 

1.0 

13 

1.0542 

1.2370 

1.055 

1.238 

1.056 

1.239 

1.057 

1.2404 

1.0576 

1.241 

14 

1.107 

1.507 

1.1088 

1.509 

1.1105 

1.511 

1.1131 

1.515 

1.1139 

1.516 

15 

1.159 

1.810 

1.1613 

1.814 

1.1639 

1.819 

1.1678 

1.825 

.1691 

1.827 

16 

1.209 

2.150 

1.2126 

2.156 

1.2161 

2.162 

1.2213 

2.171 

.2239 

2.175 

18 

1.307 

2.941 

.312 

2.952 

1.317 

2.964 

1.325 

2.983 

.328 

2.988 

20 

1.401 

3.892 

.408 

3.912 

1.415 

3.931 

1.426 

3.961 

.430 

3.971 

22 

1.492 

5.012 

.501 

5.045 

1.511 

5.075 

1.524 

5.122 

.528 

5.138 

24 

1.580 

6.321 

.591 

6.364 

.602 

6.409 

1.619 

6.475 

.625 

6.498 

26 

1.666 

7.820 

.679 

7.881 

.692 

7.942 

1.711 

8.035 

.718 

8.066 

28 

1  .  749 

9.524 

.764 

9.605 

.780 

9.687 

1.802 

9.810 

1.810 

9.852 

30 

1.831 

11.443 

.848 

11.547 

.866 

11.552 

1.890 

11.815 

1.899 

11.870 

32 

1.910 

13.585 

.929 

13.720 

.948 

13.854 

1.977 

14.06 

1.987 

14.128 

34 

1.988 

15.96 

2.009 

16.13 

2.030 

16.30 

2.062 

16.56 

2.073 

16.64 

36 

2.065 

18.59 

2.088 

18.79 

2.111 

19.0 

2.146 

19.31 

2.158 

19.42 

38 

2.140 

21.46 

2.165 

21.71 

2.190 

21.97 

2.214 

22.36 

2.240 

22.46 

146 


THE   FLOW   OF  WATER 


TABLE  D.  — Continued. 


|  Diameter  1 
|  in  Inches. 

d°-ee 

^2.66 

^0.67 

^2.67 

d°'68 

^2.68 

d  0.€95 

d  2-e95 

d0'1 

d2'7 

40 

2.214 

24.60 

2.241 

24.90 

2.268 

25.20 

2.316 

25.67 

2.324 

25.81 

42 

2.286 

28.0 

2.315 

28.36 

2.344 

28.71 

2.388 

29.26 

2.404 

29.44 

44 

2.358 

31.63 

2.388 

32.11 

2.419 

32.53 

2.467 

33.17 

2.483 

33.39 

46 

2.427 

35.67 

2.460 

36.15 

2.493 

36.63 

2.544 

37.38 

2.561 

37.63 

48 

2.496 

39.25 

2.531 

40.51 

2.567 

41.07 

2.621 

41.93 

2.038 

42.22 

50 

2.565 

44.53 

2.612 

45.17 

2.639 

45.82 

2.696 

46.87 

2.795 

47.15 

52 

2.632 

49.43 

2.671 

50.16 

2.711 

50.89 

2.771 

52.03 

2.791 

52.41 

54 

2.698 

59.65 

2.739 

55.47 

2.781 

56.32 

2.844 

57.60 

2.866 

58.03 

56 

2.764 

60.19 

2.807 

61.13 

2.851 

62.08 

2.917 

63.53 

2.940 

64.02 

58 

2.829 

66.08 

2.874 

67.13 

2.919 

68.20 

2.989 

69.83 

3.013 

70.38 

60 

2.893 

72.32 

2.940 

73.50 

2.988 

74.69 

3.060 

76.51 

3.085 

77.13 

62 

2.956 

78.91 

3.005 

80.22 

3.055 

81.55 

3.131 

83.58 

3.157 

84.26 

64 

3.019 

82.0 

3.070 

83.38 

3.121 

84.80 

3.201 

86.95 

3.228 

87.68 

66 

3.081 

91.07 

3.124 

94.79 

3.187 

96.42 

3.270 

98.92 

3.298 

99.76 

68 

3.143 

100.92 

3.198 

102.78 

3.253 

104.48 

3.339 

107.23 

3.368 

108.16 

70 

3.203 

108.98 

3.260 

110.91 

3.318 

112.89 

3.408 

115.92 

3.438 

116.94 

72 

3.262 

117.46 

3.322 

119.58 

3.382 

121.75 

3.444 

125.05 

3.505 

126.18 

78 

3.440 

145.3 

3.505 

148.0 

3.571 

150.9 

3.673 

155.1 

3.707 

186.6 

84 

3.612 

177.0 

3.683 

180.5 

3.755 

184.0 

3.867 

189.5 

3.904 

191.3 

90 

3.780 

212.7 

3.857 

217.0 

3.936 

221  A 

4.057 

228.2 

4.098 

230.5 

96 

3.945 

252.5 

4.028 

258.0 

4.112 

263.2 

4.246 

271.5 

4.287 

274.4 

102 

4.106 

296.7 

4.195 

303.1 

4.286 

309.6 

4.425 

319.7 

4.473 

323.2 

108 

4.264 

345.3 

4.359 

353.1 

4.455 

360.9 

4.604 

393.0 

4.655 

377.1 

114 

4.318 

399.7 

4.519 

407.9 

4.622 

417.1 

4.781 

431.5 

4.835 

436.4 

120 

4.571 

457.1 

4.678 

467.8 

4.786 

478.6 

4.955 

495.5 

5.011 

501.1 

126 

4.720 

520.4 

4.832 

532.8 

4.947 

545.5 

5.125 

565.0 

5.186 

571.8 

132 

4.868 

589.0 

4.986 

604.7 

5.107 

619.3 

5.294 

642.0 

5.356 

648.2 

138 

5.012 

662.9 

5.136 

679.0 

5.263 

696.1 

5.460 

722.1 

5.527 

730.9 

144 

5.155 

742.4 

5.265 

761.0 

5.418 

780.0 

5.624 

809.8 

5.694 

819.6 

150 

5.296 

827.5 

5.432 

848.7 

5.576 

870.4 

5.786 

904.0 

5.859 

915.5 

156 

5.423 

916.4 

5.564 

940.0 

5.708 

964.6 

5.931 

1002.5 

6.008 

1005.4 

162 

5.572 

1015.5 

5.719 

1042.3 

5.870 

1069.8 

6.103 

1112.5 

6.184 

1125.1 

168 

5.708 

1118.7 

5.861 

1148.6 

6.017 

1179.3 

6.260 

1226.9 

6.343 

1243.2 

174 

5.842 

1228.1 

6.0 

1261.0 

6.162 

1259.6 

6.264 

1348.6 

6.501 

1366.6 

180 

5.974 

1344.0 

6.137 

1381.0 

6.306 

1419.6 

6.567 

1478.0 

6.656 

1497.8 

186 

6.104 

1467 

6.274 

1507 

6.448 

1549 

6.716 

1614 

6.808 

1636 

192 

6.233 

1596 

6.409 

1641 

6.589 

1687 

6.869 

1758 

6.964 

1783 

198 

6.361 

1732 

6.542 

1781 

6.728 

1831 

7.017 

1910 

7.116 

1937 

204 

6.488 

1875 

6.675 

1939 

6.867 

1985 

7.164 

2071 

7.266 

2100 

210 

6.613 

2025 

6.815 

2084 

7.003 

2145 

7.310 

2238 

7.415 

2291 

216 

6.737 

2183 

6.935 

2247 

7.138 

2313 

7.454 

2415 

7.563 

2450 

222 

6.86 

2348 

7.064 

2417 

7.272 

2489 

7.597 

2600 

7.710 

2638 

228 

6.982 

2579 

7.191 

2596 

7.466 

2673 

7.740 

2794 

7.855 

2836 

234 

7.103 

2700 

7.317 

2782 

7.538 

2866 

7.881 

2997 

7.999 

3041 

240 

7.222 

2889 

7.442 

2977 

7.668 

3067 

8.021 

3208 

8.142 

3256 

OPEN   CONDUITS 


147 


TABLE  E. 

POWERS  OP  DEPTHS  OF  WATER  IN  THE  FORM  OF  SECTION  MOST  FAVOR- 
ABLE TO  FLOW.     POWERS  OF  MEAN  HYDRAULIC  RADH  IN  GENERAL. 


d  or 
r 
in 
Feet. 

m  = 
0.95 

m  = 
0.83 

m  = 
0.70 

m  = 
0.57 

m  — 
0.30 

m  = 
0.0 

K  = 
1.25 

K  = 
1  .5 

K  = 
2.00 

£O.W 

£0.68 

£0.69 

£0.70 

£0.735 

£0.75 

£0.765 

£0.775 

£0.785 

£0.87 

£0.68 

£0.69 

£0.70 

£0.735 

£0-75 

£0.765 

£0.775 

£0.796 

0.05 

0  .1344 

0  .1304 

0  .1265 

0  .1228 

0.1123 

0  .1057 

0.1010 

0  .0981 

0  .0924 

0.10 

0  .2138 

0  .2090 

0  .2044 

0  .1996 

0  .1841 

0  .1779 

0.1718 

0  .1699 

0  .1603 

0.15 

0  .2805 

0  .2752 

0  .2713 

0  .2650 

0  .2480 

0  .2410 

0  .2343 

0  .2298 

0  .2163 

0.20 

0  .3401 

0  .3348 

0  .3294 

0  .3241 

0  .3064 

0  .2990 

0  .2919 

0  .2872 

0  .2782 

0.25 

0  .3951 

0  .3896 

0  .3842 

0  .3789 

0  .3609 

0  .3535 

0  .3463 

0  .3415 

0  .3322 

0.30 

0  .4463 

0  .4410 

0  .4358 

0  .4305 

0  .4127 

0  .4054 

0  .3981 

0  .3933 

0  .3840 

0.35 

0  .4949 

0  .4897 

0  .4847 

0  .4796 

0  .4622 

0  .4550 

0  .4480 

0  .4432 

0  .4340 

0.40 

0  .5412 

0  .5363 

0  .5314 

0  .5266 

0.510 

0  .5030 

0  .4961 

0  .4915 

0  .4827 

0.45 

0  .5857 

0  .5810 

0  .5764 

0  .5718 

0  .5561 

0  .5495 

0  .5429 

0  .5386 

0  .5300 

0.50 

0  .6285 

0  .6241 

0  .6198 

0  .6156 

0  .6008 

0  .5946 

0  .5885 

0  .5844 

0  .5763 

0.55 

0.670 

0.666 

0.662 

0  .6580 

0  .6444 

0  .6389 

0  .6330 

0  .6292 

0  .6218 

0.60 

0  .7102 

0  .7066 

0  .7030 

0  .6984 

0  .6870 

0  .6818 

0  .6765 

0  .6729 

0  .6662 

0.65 

0  .7493 

0  .7461 

0  .7429 

0  .7397 

0  .7286 

0  .7239 

0  .7192 

0  .7162 

0  .7100 

0.70 

0  .7895 

0  .7846 

0  .7818 

0  .7790 

0  .7694 

0  .7653 

0  .7612 

0  .7585 

0  .7551 

0.75 

0  .8247 

0  .8223 

0  .8199 

0  .8176 

0  .8094 

0  .8059 

0  .8025 

0  .8001 

0  .7956 

0.80 

0.8611 

0  .8592 

0  .8573 

0  .8553 

0  .8487 

0  .8489 

0  .8431 

0  .8411 

0  .8374 

0.85 

0  .8968 

0  .8955 

0  .8939 

0  .8924 

0  .8874 

0  .8853 

0.8831 

0  .8815 

0  .8788 

0.90 

0  .9318 

0  .9309 

0.930 

0  .9289 

0  .9254 

0  .9246 

0  .9226 

0  .9216 

0  .9196 

0.95 

0  .9662 

0  .9657 

0  .9652 

0  .9647 

0  .9630 

0  .9623 

0  .9615 

0  .9610 

0  .9600 

1.0 

.0 

1  .0 

1  .0 

1.0 

.0 

1  .0 

1  .0 

1  .0 

1.0 

1  .05 

.0332 

1  .0337 

1  .0342 

1  .0347 

.0365 

1  .0373 

1  .0380 

1  .0385 

1  .0396 

1.10 

.0660 

.067 

1  .068 

1  .0690 

.0726 

1  .0741 

1  .0756 

.0768 

1  .0787 

1  .15 

.0982 

.0997 

1  .1012 

1  .1028 

.1082 

1  .1105 

1  .1128 

.1144 

1  .116 

1.20 

.130 

.132 

1.134 

1  .1361 

.1434 

1  .1465 

1  .1497 

.1518 

1  .156 

1.25 

.1612 

.1638 

1  .1664 

1  .1691 

.1782 

1  .1782 

1  .1861 

.1888 

1  .1941 

.30 

.192 

.195 

1.199 

1  .2016 

.2126 

1  .2174 

1  .2222 

.225 

1  .2319 

.35 

.223 

.227 

1.230 

1  .2338 

.2468 

1  .252 

1  .258 

.262 

1.269 

.40 

.244 

.248 

1  .253 

1  .265 

.281 

1  .287 

1  .294 

.298 

1.307 

.45 

1  .283 

.287 

1  .292 

1  .297 

.314 

1  .321 

1  .329 

.304 

1  .343 

.50 

1  .312 

.318 

1  .323 

1.328 

.347 

1  .355 

1  .364 

.369 

1  .380 

1  .55 

1  .341 

.347 

1  .353 

1  .359 

.380 

1  .389 

1  .398 

.405 

1  .417 

1  .60 

1  .370 

.377 

1.383 

1  .389 

.413 

1.423 

1  .433 

.440 

1  .453 

1.65 

1  .399 

.406 

1.413 

1.420 

.445 

1.456 

1.467 

.474 

1  .489 

1.70 

1.427 

.435 

1  .442 

1.450 

.477 

1.489 

1  .501 

.509 

1  .525 

1.75 

1.455 

.463 

1.471 

1  .480 

.509 

1  .521 

1  .534 

.543 

1  .561 

1.80 

1  .483 

.491 

1.500 

1.509 

.540 

1  .554 

1  .568 

.577 

1  .595 

1  .85 

1  .510 

.520 

1  .529 

1  .538 

.572 

1  .586 

1  .601 

1  .611 

1.631 

1  .90 

1  .537 

.547 

1  .557 

1  .569 

.603 

1  .618 

1  .634 

1  .644 

1.665 

1  .95 

.564 

.575 

1  .586 

1  .596 

.634 

1  .650 

1  .667 

1  .717 

1  .699 

2.0 

.589 

.600 

1  .611 

1.624 

.664 

1  .681 

1  .700 

1  .711 

1  734 

2.05 

.618 

.629 

1.641 

1.653 

.695 

1  .713 

1  .732 

1  .744 

1  .769 

2.10 

.644 

.657 

1  .669 

1.681 

.725 

1.745 

1  .764 

1.777 

1.804 

2.15 

.670 

.683 

1.696 

1.709 

.755 

1.776 

1.796 

1.810 

1.837 

2.20 

.696 

.710 

1  .723 

1  .736 

.785 

1.809 

1.828 

1.842 

1.861 

148 


THE   FLOW   OF    WATER 


TABLE   E.  —  Continued. 


d  or 

7» 

m  = 
0.95 

m  = 
0.83 

m  = 
0.70 

tn  — 
0.57 

m  = 
0.30 

m  = 
0.0 

K  = 
1.25 

K  = 
1  .50 

7^-     

2.00 

in 
fppt 

£0.67 

£0-68 

£0.69 

£0-70 

£0.735 

£0.75 

£0.765 

£0.775 

£0.795 

H  I  •  I  . 

£0.67 

£0.68 

£0.69 

£0.70 

£0.735 

£0.75 

£0.765 

£0.775 

£0.795 

2.25 

1  .718 

.732 

1  .746 

1  .764 

1  .815 

1   .836 

1  .860 

1   .875 

1.905 

2.30 

.747 

.762 

1.777 

1  .792 

1  .844 

1  .867 

1  .891 

1  .907 

1.939 

2.35 

.773 

.788 

.803 

1.818 

1.874 

1  .898 

1  .922 

1  .939 

1  .973 

2.40 

.798 

.814 

.830 

1.846 

1  .903 

1.928 

1  .954 

1  .971 

2.006 

2.45 

.826 

.839 

.856 

1.872 

1.932 

1.958 

1  .985 

2.003 

2.038 

2.50 

.848 

1.864 

.882 

1.899 

1.961 

1  .960 

2.016 

2.034 

2.069 

2.55 

1.872 

1.890 

.908 

1.926 

1.990 

2.018 

2.046 

2.065 

2.104 

2.60 

1  .897 

1.915 

.934 

1.952 

2.019 

2.047 

2.077 

2.097 

2.137 

2.65 

1.921 

1.940 

.959 

1  .978 

2.048 

2.077 

2.108 

2.138 

2.169 

2.70 

1  .946 

1  .965 

.984 

2.004 

2.075 

2.106 

2.136 

2.160 

2.203 

2.75 

1  .970 

1  .990 

2.010 

2.030 

2.103 

2.135 

2.168 

2.190 

2.212 

2.80 

1  .994 

2.014 

2.035 

2.056 

2.131 

2.165 

2.198 

2.221 

2.267 

2.85 

2.017 

2.038 

2.060 

2.081 

2.159 

2.194 

2.228 

2.252 

2.306 

2.90 

2.041 

2.063 

2.085 

2.107 

2.187 

2.223 

2.258 

2.282 

2.331 

2.95 

2.064 

2.089 

2.110 

2.132 

2.215 

2.251 

2.288 

2.313 

2.363 

3.0 

2.088 

2.111 

2.134 

2.157 

2.242 

2.279 

2.317 

2.343 

2.395 

3.05 

2.111 

2.135 

2.159 

2.183 

2.270 

2.308 

2.347 

2.373 

2.426 

3.10 

2.134 

2.159 

2.183 

2.208 

2.297 

2.337 

2.376 

2.403 

2.458 

3.15 

2.159 

2.182 

2.207 

2.233 

2.324 

2.364 

2.406 

2.433 

2.488 

3.20 

2.180 

2.205 

2.231 

2.258 

2.351 

2  ,393 

2.435 

2.463 

2.521 

3.25 

2.201 

2.229 

2.255 

2.282 

2.378 

2.421 

2.464 

2.493 

2.551 

3.30 

2.225 

2.252 

2.279 

2.307 

2.421 

2.448 

2.493 

2.522 

2.582 

3.35 

2.248 

2.275 

2.303 

2.331 

2.432 

2.476 

2.521 

2.552 

2.614 

3.40 

2.270 

2.298 

2.327 

2.355 

2.458 

2.504 

2.550 

2.580 

2.645 

3.45 

2.293 

2.323 

2.354 

2.379 

2.485 

2.531 

2.579 

2.611 

2.677 

3.50 

2.315 

2.344 

2.374 

2.404 

2.511 

2.559 

2.608 

2.640 

2.707 

3.55 

2.337 

2  .367 

2.397 

2.428 

2.538 

2.586 

2.636 

2.666 

2.738 

3.60 

2.359 

2.389 

2.420 

2.451 

2.564 

2.613 

2.664 

2.699 

2.769 

3.65 

2.381 

2.412 

2.443 

2.476 

2.590 

2.641 

2.693 

2.725 

2.799 

3.70 

2.403 

2.434 

2.467 

2.499 

2.616 

2.668 

2.720 

2.755 

2.830 

3.75 

2.424 

2.457 

2.489 

2.522 

2.642 

2.695 

2.749 

2.785 

2.858 

3.80 

2.446 

2.479 

2.512 

2.546 

2.668 

2.722 

2.777 

2.814 

2.890 

3.85 

2.468 

2.501 

2.536 

2.590 

2.693 

2.748 

2.804 

2.843 

2.920 

3.90 

2.489 

2  .523 

2.558 

2.593 

2.719 

2.775 

2.833 

2.871 

2.951 

3.95 

2.510 

2.545 

2.580 

2.616 

2.745 

2.802 

2.860 

2.900 

2.980 

4.0 

2.531 

2.567 

2.603 

2.639 

2.770 

2.828 

2.888 

2.928 

3.01 

4.05 

2.552 

2.588 

2.626 

2.662 

2.796 

2.855 

2.915 

2.956 

3.04 

4.10 

2.573 

2.610 

2.647 

2.685 

2.821 

2.881 

2.943 

2.985 

3.07 

4.15 

2.595 

2.632 

2.670 

2.708 

2.846 

2.908 

2.970 

3.013 

3.10 

4.20 

2.616 

2.653 

2.692 

2.731 

2.871 

2.934 

2.998 

3.041 

3.13 

4.25 

2.637 

2.675 

2.714 

2.753 

2.897 

2.960 

3.025 

3.070 

3.16 

4.30 

2.657 

2.696 

2.736 

2.776 

2.921 

2.986 

3.052 

3.097 

3.19 

4.35 

2.678 

2.717 

2.758 

2.798 

2.946 

3.012 

3.080 

3.125 

3.219 

4.40 

2.698 

2.739 

2.780 

2.821 

2.971 

3.038 

3.107 

3.153 

3.247 

4.45 

2.719 

2.760 

2.801 

2.844 

2.997 

3.064 

3.134 

3.180 

3.277 

4.50 

2.740 

2.781 

2.823 

2.866 

3.021 

3.090 

3.160 

3.208 

3.306 

4.55 

2.760 

2.802 

2.845 

2.888 

3.045 

3.115 

3.187 

3.235 

3.335 

4.60 

2.780 

2.823 

2.866 

2.910 

3.070 

3.140 

3.214 

3.263 

3.364 

OPEN    CONDUITS 


149 


TABLE   E.  —  Continued. 


d  or 
r 
in 
fppt 

m  = 
0.95 

m  = 
0.83 

m  = 
0.70 

m  = 
0.57 

m  = 
0.30 

m  = 
0.0 

K  = 
1.25 

K  = 
1.50 

K  = 
20 

£)0.67 

£0.68 

£0.69 

£0.70 

£0.735 

£0.75 

£0.765 

£0.775 

£0-705 

It?"  t. 

£0-67 

£0.68 

£0.<59 

£0.70 

£0.735 

£0.75 

£0.765 

£0.775 

£0-795 

4.65 

2.800 

2.844 

2.888 

2.932 

3.094 

3.166 

3.240 

a.  290 

3.393 

4.70 

2.820 

2.864 

2.909 

2.954 

3.119 

3.192 

3.267 

3  .318 

3.422 

4.75 

2.840 

2.885 

2.930 

2.976 

3.143 

3.218 

3.294 

3.345 

3.451 

4.80 

2.860 

2.906 

2.952 

2.998 

3.163 

3.243 

3.320 

3  .373 

3.480 

4.85 

2.880 

2.926 

2.973 

3.020 

3.192 

3.268 

3  .346 

3.404 

3.509 

4.90 

2.900 

2.947 

2.994 

3.042 

3.216 

3.295 

3.373 

3.427 

3.538 

4.95 

2.920 

2.967 

3.015 

3.064 

3.240 

3.318 

3.397 

3  .454 

3.567 

5.00 

2.940 

2.987 

3.036 

3.085 

3.264 

3.344 

3.425 

3.481 

3.595 

5.1 

2.980 

3.028 

3.078 

3.128 

3.312 

3.393 

3.478 

3.534 

3.652 

5.2 

3.018 

3.068 

3.119 

3.171 

3.360 

3.444 

3.530 

3.589 

3.709 

5.3 

3.057 

3.108 

3.160 

3.214 

3.407 

3.493 

3.582 

3.642 

3.765 

5.4 

3.095 

3.148 

3.201 

3.256 

3.454 

3.542 

3.633 

3.695 

3.822 

5.5 

3.134 

3.188 

3.243 

3.298 

3.501 

3.591 

3.684 

3.748 

3.878 

5.6 

3.171 

3.227 

3.283 

3.340 

3.548 

3.640 

3.735 

3.818 

3.933 

5.7 

3.210 

3.266 

3.323 

3.382 

3.594 

3.689 

3.787 

3.853 

3.990 

5.8 

3.249 

3.307 

3.365 

3.423 

3.640 

3.737 

3.837 

3.908 

4.045 

5.9 

3.284 

3.343 

3.403 

3.464 

3.686 

3.786 

3.888 

3.958 

4.101 

6.0 

3.322 

3.382 

3.443 

3.505 

3.732 

3.834 

3.938 

4.009 

4.156 

6.1 

3.358 

3.420 

3.482 

3.546 

3.778 

3.882 

3.988 

4.061 

4.211 

6.2 

3.396 

3.458 

3.521 

3.586 

3.823 

3.929 

4.038 

4.113 

4.265 

6.3 

3.432 

3.496 

3.561 

3.627 

3.868 

3.977 

4.087 

4.164 

4.320 

6.4 

3.469 

3.534 

.3.600 

3.667 

3.913 

4.604 

4.138 

4.215 

4.375 

6.5 

3.505 

3.571 

3.638 

3.707 

3.958 

4.071 

4.187 

4.265 

4.429 

6.6 

3.541 

3.608 

3.677 

3.747 

4.003 

4.118 

4.236 

4.317 

4.483 

6.7 

3.576 

3.645 

3.715 

3.787 

4.047 

4.164 

4.285 

4.367 

4.537 

6.8 

3.612 

3.682 

3.754 

3.826 

4.091 

4.211 

4.334 

4.417 

4.590 

6.9 

3.648 

3.719 

3.792 

3.865 

4.136 

4.257 

4.383 

4.468 

4.644 

7.0 

3.683 

3.755 

3.829 

3.905 

4.180 

4.304 

4.431 

4.518 

4.698 

7.1 

3.718 

3.792 

3.867 

3.944 

4.223 

4.350 

4.480 

4.567 

4.751 

7.2 

3.753 

3.828 

3.904 

3.982 

4.267 

4.396 

4.528 

4.618 

4.804 

7.3 

3.788 

3.864 

3.942 

4.021 

4.311 

4.441 

4.576 

4.667 

4.857 

7.4 

3.822 

3.900 

3.979 

4.069 

4.354 

4.487 

4.623 

4.717 

4.910 

7.5 

3.857 

3.936 

4.016 

4.097 

4.397 

4.532 

4.671 

4.766 

4.962 

7.6 

3.892 

3.972 

4.053 

4.136 

4.439 

4.577 

4.719 

4.816 

5.014 

7.7 

3.926 

4.007 

4.090 

4.174 

4.483 

4.622 

4.766 

4.865 

5.066 

7.8 

3.960 

4.042 

4.126 

4.212 

4.526 

4.667 

4.813 

4.913 

5.119 

7.9 

3.994 

4.077 

4.163 

4.250 

4.568 

4.710 

4.860 

4.962 

5.171 

8.0 

4.028 

4.113 

4.199 

4.278 

4.611 

4.754 

4.908 

5.010 

5.223 

8.1 

4.061 

4.147 

4.235 

4.325 

4.653 

4.801 

4.954 

5.059 

5.275 

8.2 

4.095 

4.182 

4.271 

4.362 

4.696 

4.847 

5.002 

5.108 

5.328 

8.3 

4.128 

4.217 

4.306 

4.399 

4.737 

4.890 

5.047 

5.155 

5.379 

8.4 

4.162 

4.251 

4.343 

4.436 

4.779 

4.934 

5.094 

5.204 

5.430 

8.5 

4.195 

4.286 

4.378 

4.473 

4.821 

4.978 

5.141 

5.251 

5.481 

8.6 

4.228 

4.320 

4.414 

4.516 

4.862 

5.022 

5.184 

5.300 

5.532 

8.7 

4.261 

4.354 

4.449 

4.547 

4.904 

5.066 

5.233 

5.348 

5.584 

8.8 

4.294 

4.388 

4.484 

4.583 

4.945 

5.109 

5.279 

5.395 

5.635 

8.9 

4.326 

4.422 

4.519 

4.619 

4.987 

5.153 

5.324 

5.442 

5.686 

9.0 

4.359 

4.455 

4.555 

4.656 

5.028 

5.196 

5.370 

5.490 

5.736 

150 


THE  FLOW   OF   WATER 


TABLE   E.  —  Concluded. 


d  or 
r  in 
Feet. 

m  = 
0.95 

1YI  == 

0.83 

m  = 
0.70 

m  = 
0.57 

m  = 
0.30 

m  = 
0.0 

K  = 
1.25 

K  = 
1.50 

K  = 
2.00 

£0.67 

£0.68 

£0.69 

£0.70 

£0.735 

£0.75 

£0.765 

£0.775 

£0.795 

£0.«7 

£0-6* 

£0.09 

£0-70 

£0.735 

£0.75 

£0.765 

£0.775 

£0.796 

9.1 

4.391 

4.489 

4.589 

4.692 

5.069 

5.239 

5.416 

5.537 

5.786 

9.2 

4.423 

4.522 

4.624 

4.728 

5.110 

5.282 

5.461 

5.584 

5.837 

9.3 

4.455 

4.556 

4.659 

4.764 

5.150 

5.325 

5.506 

5.631 

5.888 

9.4 

4.487 

4.589 

4.693 

4.799 

5.191 

5.368 

5.552 

5.678 

5.938 

9.5 

4.519 

4.622 

4.728 

4.835 

5.232 

5.411 

5.597 

5.725 

5.989 

9.6 

4.551 

4.655 

4.762 

4.871 

5.272 

5.454 

5.642 

5.772 

6.039 

9.7 

4.583 

4.688 

4.796 

4.906 

5.312 

5.496 

5.687 

5.818 

6.089 

9.8 

4.615 

4.721 

4.831 

4.942 

5.352 

5.539 

5.732 

5.864 

6.138 

9.9 

4.646 

4.754 

4.865 

4.977 

5.393 

5.582 

5.779 

5.910 

6.188 

10.0 

4.678 

4.787 

4.898 

5.012 

5.433 

5.624 

5.821 

5.957 

6.238 

10.5 

4.833 

4.948 

5.065 

5.186 

5.631 

5.833 

6.042 

6.186 

6.484 

11.0 

4.986 

5.107 

5.231 

5.358 

5.827 

6.040 

6.261 

6.413 

6.728 

11.5 

5.137 

5.263 

5.394 

5.527 

6.020 

6.245 

6.478 

6.638 

6.969 

12.0 

5.285 

5.418 

5.554 

5.695 

6.211 

6.447 

6.692 

6.861 

7.210 

12.5 

5.432 

5.570 

5.713 

5.859 

6.401 

6.648 

6.905 

7.082 

7.437 

13.0 

5.576 

5.721 

5.870 

6.022 

6.588 

6.846 

7.115 

7.300 

7.684 

13.5 

5.719 

5.870 

6.025 

6.183 

6.773 

7.043 

7.323 

7.516 

7.907 

14.0 

5.860 

6.017 

6.178 

6.343 

6.957 

7.238 

7.530 

7.731 

8.150 

14.5 

6.000 

6.162 

6.329 

6.501 

7.139 

7.430 

7.735 

7.945 

8.380 

15.0 

6.137 

6.306 

6.479 

6.657 

7.319 

7.622 

7.938 

8.156 

8.610 

15.5 

6.274 

6.448 

6.627 

.6.812 

7.497 

7.812 

8.140 

8.369 

8.837 

16.0 

6.409 

6.589 

6.777 

6.964 

7.674 

8.000 

8.340 

8.574 

9.063 

16.5 

6.542 

6.728 

7.016 

7.116 

7.850 

8.187 

8.538 

8.782 

9.287 

17.0 

6.674 

6.866 

7.064 

7.266 

8.024 

8.372 

8.736 

8.986 

9.511 

17.5 

6.805 

7.003 

7.206 

7.415 

8.197 

8.556 

8.932 

9.191 

9.732 

18.0 

6.935 

7.138 

7.347 

7.563 

8.368 

8.739 

9.125 

9.395 

9.953 

18.5 

7.063 

7.272 

7.488 

7.709 

8.538 

8.920 

9.320 

9.596 

10.172 

19.0 

7.191 

7.405 

7.626 

7.855 

8.707 

9.101 

9.512 

9.796 

10.390 

19.5 

7.317 

7.538 

7.765 

7.999 

8.875 

9.280 

9.702 

9.996 

10.610 

20.0 

7.442 

7.668 

7.902 

8.142 

9.042 

9.410 

9.892 

10.192 

10.830 

21.0 

7.689 

7.927 

8.172 

8.425 

9.372 

9.810 

10.268 

10.585 

11.25 

22.0 

7.806 

8.182 

8.439 

8.704 

9.698 

10.168 

10.640 

10.974 

11.674 

23.0 

8.173 

8.433 

8.702 

8.979 

10.020 

10.502 

11.008 

11.359 

12.094 

24.0 

8.409 

8.680 

8.961 

9.250 

10.338 

10.843 

11.373 

11.739 

12.510 

25.0 

8.642 

8.925 

9.217 

9.518 

10.653 

11.180 

11.734 

12.117 

12.920 

OPEN   CONDUITS 


151 


TABLE  F. 

POWERS  OP  THE  DEPTHS  OF  WATER  IN  THE  FORM  OF  SECTION  MOST 
FAVORABLE  TO  FLOW. 


d  in 
Feet. 

m  = 
0.95 

m  = 
0.83 

tn= 
0.70 

m  = 
0.57 

m  = 
0.30 

m  = 
0.0 

K= 
1.25 

K= 
1.5 

X= 

2.0 

d2.795 

d2'67 

d2-68 

d2.69 

d2'70 

d2-735 

d2'75 

d2-765 

d2'775 

0.05 

.000336 

.000326 

.000316 

.000307 

.000281 

.000264 

.000252 

.000245 

.000231 

0.10 

.00214 

.00209 

.00204 

.00200 

.001841 

.00178 

.001919 

.00168 

.001604 

0.15 

.00631 

.00619 

.00610 

.00596 

.00558 

.00542 

.00527 

.00517 

.00428 

0.20 

.01361 

.01339 

.01311 

.01292 

.01226 

.01196 

.01168 

.01150 

.01113 

0.25 

.02469 

.02435 

.02401 

.02368 

.02256 

.02210 

.02164 

.02134 

.02076 

0.30 

.04017 

.03969 

.03921 

.03875 

.03715 

.03648 

.03583 

.03540 

.03456 

0.35 

.06063 

.06000 

.05936 

.05875 

.05663 

.05574 

.05488 

.05430 

.05317 

0.40 

.08660 

.08581 

.08502 

.08425 

.08159 

.08048 

.07939 

.07866 

.07723 

0.45 

.1186 

.1176 

.1167 

.1158 

.1126 

.1113 

.1099 

.1091 

.1093 

0.50 

.1571 

.1560 

.1550 

.1534 

.1502 

.1487 

.1471 

.1461 

.1441 

0.55 

.2026 

.2015 

.2003 

.1990 

.1949 

.1932 

.1914 

.1903 

.1881 

0.60 

.2557 

.2544 

.2531 

.2518 

.2473 

.2454 

.2435 

.2421 

.2398 

0.65 

.3163 

.3152 

.3139 

.3125 

.3078 

.3058 

.3039 

.3026 

.3000 

0.70 

.3859 

.3845 

.3831 

.3817 

.3770 

.3750 

.3730 

.3717 

.3690 

0.75 

.4639 

.4626 

.4613 

.4600 

.4553 

.4533 

.4514 

.4501 

.4475 

0.80 

.5511 

.5428 

.5487 

.5475 

.5432 

.5413 

.5396 

.5383 

.5360 

0.85 

.6480 

.6469 

.6459 

.6449 

.6412 

.6396 

.6380 

.6370 

.6345 

0.90 

.7552 

.7542 

.7532 

.7522 

.7496 

.7484 

.7472 

.7465 

.7450 

0.95 

.8720 

.8716 

.8711 

.8707 

.8691 

.8684 

.8678 

.8672 

.8664 

1.0 

1.0 

1.0 

1.0 

1.0 

.0 

1.0 

.0 

1.0 

1.0 

1.05 

1.1391 

1.1397 

1.1402 

1.1408 

.1428 

1.1436 

.1444 

1  .  1450 

1.1461 

1.10 

1.290 

1.291 

1.292 

1.294 

.298 

1.300 

.302 

1.303 

1.305 

1.15 

1.452 

1.454 

1.456 

1.458 

.465 

1.469 

.472 

1.474 

1.478 

1.20 

1.627 

1.630 

1.633 

1.636 

.647 

1.651 

.656 

1.659 

1.665 

1.25 

1.814 

1.819 

1.823 

1.827 

1.841 

1.847 

1.853 

1.858 

1.866 

1.30 

2.015 

2.020 

2.025 

2.030 

2.050 

2.059 

2.066 

2.071 

2.082 

1.35 

2.228 

2.235 

2.242 

2.249 

2.272 

2.283 

2.293 

2.30 

2.314 

1.40 

2.456 

2.464 

2.472 

2.481 

2.510 

2.523 

2.535 

2.544 

2.561 

.45 

2.697 

2.707 

2.717 

2.727 

2.763 

2.778 

2.794 

2.804 

2.825 

.50 

2.952 

2.964 

2.976 

2.988 

3.031 

3.050 

3.068 

3.081 

3.106 

.55 

3.222 

3.236 

3.251 

3.265 

3.316 

3.338 

3.359 

3.377 

3.404 

.60 

3.508 

3.524 

3.540 

3.557 

3.616 

3.641 

6.668 

3.685 

3.720 

.65 

3.808 

3.827 

3.846 

3.866 

3.934 

3.954 

3.994 

4.014 

4.054 

.70 

4.124 

4.146 

4.168 

4.190 

4.268 

4.303 

4.337 

4.360 

4.407 

.75 

4.456 

4.481 

4.506 

4.531 

4.621 

4.660 

4.699 

4.726 

4.778 

.80 

4.804 

4.832 

4.860 

4.889 

4.991 

5.035 

5.080 

5.110 

5.170 

.85 

5.168 

5.200 

5.232 

5.265 

5.379 

5.429 

5.479 

5.513 

5.582 

.90 

5.550 

5.585 

5.621 

5.658 

5.786 

5.842 

5.899 

5.937 

6.013 

.95 

5.962 

6.002 

6.042 

6.068 

6.212 

6.274 

6.338 

6.380 

6.466 

2.0 

6.356 

6.400 

6.444 

6.498 

6.643 

6.727 

6.797 

6.845 

6.940 

2.05 

6.798 

6.847 

6.896 

6.947 

7.123 

7.198 

7.278 

7.330 

7.436 

2.10 

7.250 

7.304 

7.358 

7.413 

7.608 

7.693 

7.780 

7.837 

7.955 

2.15 

7.720 

7.779 

7.839 

7.899 

8.114 

8.208 

8.302 

8.366 

8.495 

2.20 

8.209 

8.273 

8.339 

8.405 

8.640 

8.743 

8.847 

8.917 

9.059 

2.25 

8.696 

8.767 

8.838 

8.931 

9.188 

9.301 

9.415 

9.464 

9.646 

2.30 

9.243 

9.321 

9.398 

9.477 

9.757 

9.880 

10.004 

10.111 

10.256 

2.35 

9.790 

9.873 

9.958 

10.044 

10.348 

10.482 

10.616 

10.708 

10.892 

152 


THE    FLOW   OF   WATER 
TABLE   F. —  Continued. 


d  in 
Feet 

TO  = 

0.95 

m  = 
0.83 

d2-68 

7tt  = 

0.70 

m  = 
0.57 

m  = 
0.30 

7«  = 

0.0 

K  = 
1.25 

K  = 
1.5 

K  = 
2.0 

d2'67 

^2-69 

d2.70 

^2.735 

d2-75 

d2-765 

^2.775 

rf2-795 

2.40 

10.355 

10.446 

10.538 

10.631 

10.952 

11.106 

11.253 

11.352 

11.553 

2.45 

10.141 

11.040 

11.139 

11.265 

11.597 

11.754 

11.913 

12.021 

12.276 

2.50 

11.548 

11.654 

11.761 

11.870 

12.256 

12.426 

12.600 

12.710 

12.950 

2.55 

12.175 

12.289 

12.405 

12.52 

12.92 

13.12 

13.21 

13.43 

13.69 

2.60 

12.82 

12.95 

13.07 

13.20 

13.34 

13.84 

14.04 

14.18 

14.45 

2.65 

13.49 

13.63 

13.76 

13.89 

14.37 

14.59 

14.80 

14.94 

15.27 

2.70 

14.18 

14.32 

14.47 

14.63 

15.13 

15.35 

15.59 

15.74 

16.06 

2.75 

14.90 

15.05 

15.20 

15.35 

15.91 

16.15 

16.40 

16.56 

16.90 

2.80 

15.63 

15.79 

15.95 

16.12 

16.71 

16.97 

17.23 

17.41 

17.77 

2.85 

16.88 

16.56 

16.73 

16.91 

17.54 

17.82 

18.10 

18.29 

18.68 

2.90 

17.16 

17.35 

17.53 

17.72 

18.39 

18.69 

18.99 

19.19 

19.61 

2.95 

17.97 

18.16 

18.36 

18.56 

19.27 

19.59 

19.91 

20.12 

20.56 

3.0 

18.79 

19.00 

19.21 

19.42 

20.18 

20.51 

20.86 

21.09 

21.56 

3.05 

19.64 

19.86 

20.08 

20.31 

21.11 

21.47 

21.83 

22.08 

22.57 

3.10 

20.51 

20.75 

20.98 

21.22 

22.09 

22.45 

22.84 

23.10 

23.62 

3.15 

21.40 

21.65 

21.90 

22.15 

23.06 

23.46 

23.87 

24.14 

24.70 

3.20 

22.32 

22.58 

22.85 

23.12 

24.08 

24.50 

24.93 

25.22 

25.81 

3.25 

23.27 

23.54 

23.82 

24.10 

25.11 

25.56 

26.02 

26.33 

26.96 

3.30 

24.24 

24.53 

24.82 

25.11 

26.19 

26.67 

27.14 

27.47 

28.14 

3.35 

25.23 

25.53 

25.84 

26.16 

27.29 

27.79 

28.30 

28.64 

29.34 

3.40 

26.26 

26.59 

26.98 

27.23 

28.42 

28.95 

29.48 

29.84 

30.58 

3.45 

27.29 

27.65 

28.02 

28.32 

29.58 

30.13 

30.69 

31.08 

31.86 

3.50 

28.36 

28.72 

29.00 

29.44 

30.76 

31.34 

31.94 

32.34 

33.16 

3.55 

29.45 

29.83 

30.21 

30.59 

31.98 

32.60 

33.22 

33.60 

34.50 

3.60 

30.57 

30.97 

31.37 

31.77 

33.23 

33.87 

34.53 

34.97 

35.88 

3.65 

31.72 

32.13 

32.55 

32.97 

34.50 

35.18 

35.87 

36.34 

37.29 

3.70 

32.89 

33.33 

33.76 

34.21 

35.81 

36.52 

37.25 

37.74 

38.74 

3.75 

34.09 

34.55 

35.01 

35.47 

37.15 

37.89 

38.65 

39.17 

40.22 

3.80 

35.32 

35.80 

36.26 

36.80 

38.53 

39.30 

40.10 

40.63 

41.73 

3.85 

36.57 

37.07 

37.57 

38.08 

39.93 

40.74 

40.62 

42.14 

43.29 

3.90 

39.86 

38.37 

38.90 

39.43 

41.36 

42.21 

43.08 

43.67 

44.88 

3.95 

39.17 

39.71 

40.26 

40.82 

42.82 

43.42 

44.62 

45.24 

46.50 

4.0 

40.51 

41.07 

41.64 

42.22 

44.34 

45.25 

46.20 

46.85 

48.17 

4.05 

41.87 

42.46 

43.06 

43.67 

45.85 

46.82 

47.82 

48.49 

49.87 

4.10 

43.27 

43.88 

44.50 

45.14 

47.42 

48.43 

49.47 

50.17 

51.61 

4.15 

44.89 

45.54 

46.19 

46.64 

49.02 

50.08 

51.15 

52.13 

53.39 

4.20 

46.14 

46.81 

47.48 

48.17 

50.65 

51.76 

52.88 

53.65 

55.20 

4.25 

47.62 

48.38 

49.02 

49.74 

52.32 

53.46 

54.64 

55.46 

57.06 

4.30 

49.00 

49.85 

50.59 

51.33 

54.02 

55.21 

56.44 

57.26 

58.96 

4.35 

50.67 

51.42 

52.19 

53.06 

55.95 

57.00 

58.27 

59.12 

60.89 

4.40 

52.24 

53.02 

53.81 

54.62 

57.52 

58.81 

60.14 

61.03 

62.87 

4.45 

53.84 

54.66 

55.47 

56.31 

59.33 

60.68 

62.05 

62.98 

64.89 

4.50 

55.47 

56.31 

57.17 

58.03 

61.17 

62.57 

63.99 

64.96 

66.25 

4.55 

57.13 

58.01 

58.89 

59.79 

63.05 

64.50 

65.  C8 

66.98 

69.05 

4.60 

58.83 

59.73 

60.65 

61.58 

64.96 

66.46 

68.00 

69.05 

71.19 

4.65 

60.44 

61.37 

62.32 

63.41 

66.91 

68.47 

70.07 

71.02 

73.37 

4.70 

62.30 

63.28 

64.26 

65.26 

68.89 

70.51 

72.17 

73.29 

75.60 

4.75 

64.09 

65.09 

66.12 

67.15 

70.92 

72.76 

74.31 

75.48 

77.87 

4.80 

65.91 

66.95 

1 

68.00 

69.08 

72.98 

74.72 

76.49 

77.70 

80.18 

OPEN    CONDUITS 
TABLE   F.  —  Continued. 


153 


d  in 
Feet. 

m== 
0.95 

m= 
0.83 

m  = 
0.70 

m  = 
0.57 

m= 
0.30 

772.  — 

0.0 

K= 
1.25 

K= 
1.5 

K  = 

2.0 

d2'67 

d2-68 

d2.69 

d2'70 

d2.735 

d2-75 

ffi-™ 

^2-775 

d2-795 

4.85 

67.75 

68.83 

69.93 

71.04 

75.08 

76.88 

78.72 

80.06 

82.54 

4.90 

69.65 

70.75 

71.88 

73.05 

77.21 

79.08 

80.98 

82.28 

84.94 

4.95 

71.54 

72.70 

73.87 

75.07 

79.39 

81.31 

83.29 

84.63 

87.38 

5.0 

73.50 

74.69 

75.90 

77.00 

81.60 

83.59 

85.63 

87.03 

89.87 

5.0 

77.49 

78.77 

80.06 

81.37 

86.14 

88.27 

90.46 

91.92 

94.99 

5.2 

81.61 

82.96 

84.34 

85.25 

90.85 

93.12 

95.44 

96.85 

100.3 

5.3 

85.87 

87.31 

88.78 

90.27 

95.69 

98.12 

100.61 

102.30 

105.77 

5.4 

90.26 

91.80 

93.36 

94.94 

100.71 

103.29 

105.94 

107.89 

111.44 

5.5 

94.79 

96.43 

98.09 

99.76 

105.90 

108.67 

111.46 

113.37 

117.31 

5.6 

99.46 

101.20 

102.95 

104.74 

111.25 

114.16 

116.65 

119.74 

123.34 

5.7 

104.08 

105.80 

107.79 

109.87 

116.77 

119.85 

123.02 

125.2 

129.6 

5.8 

109.50 

111.44 

113.42 

115.15 

122.45 

125.7 

129.1 

121.7 

126.1 

5.9 

114.3 

116.4 

118.5 

120.6 

128.3 

131.8 

135.5 

137.8 

142.7 

6.0 

119.6 

121.7 

123.9 

126.2 

134.4 

138.0 

141.8 

144.3 

149.6 

6.1 

125.0 

127.2 

129.6 

132.0 

140.6 

144.4 

148.4 

150.9 

156.7 

6.2 

130.5 

132.9 

135.3 

137.9 

147.0 

151.0 

155.2 

158.1 

164.0 

6.3 

136.2 

128.8 

141.7 

144.0 

153.5 

157.8 

162.3 

165.3 

171.5 

6.4 

142.0 

144.7 

147.4 

150.2 

160.3 

164.8 

169.5 

172.6 

179.2 

6.5 

148.1 

150.9 

153.7 

156.7 

167.2 

172.0 

176.9 

180.2 

187.1 

6.6 

154.2 

157.2 

160.1 

163.2 

174.4 

179.4 

184.1 

188.0 

195.2 

6.7 

160.5 

163.5 

166.8 

170.0 

181.7 

187.0 

192.4 

195.9 

203.6 

6.8 

167.0 

170.3 

173.6 

176.9 

189.2 

194.7 

200.4 

204.3 

212.2 

6.9 

173.7 

177.1 

180.4 

184.1 

196.9 

202.7 

213.5 

212.7 

221.1 

7.0 

180.5 

184.0 

187.6 

191.3 

204.8 

210.9 

217.1 

226.5 

230.2 

7.1 

187.4 

191.1 

194:9 

198.8 

212.9 

219.2 

225.8 

230.2 

239.5 

7.2 

193.7 

197.5 

202.4 

206.5 

221.2 

227.9 

234.7 

239.4 

249.0 

7.3 

201.9 

205.9 

210.0 

214.2 

229.7 

236.7 

243.8 

248.7 

258.8 

7.4 

209.3 

213.6 

217.9 

222.3 

238.4 

245.7 

253.2 

258.3 

268.9 

7.5 

217.0 

221.4 

225.9 

230.5 

247.3 

254.9 

262.8 

268.1 

279.1 

7.6 

224.8 

229.4 

234.1 

238.9 

256.5 

264.4 

272.5 

278.1 

289.8 

7.7 

232.8 

237.6 

242.4 

247.5 

265.8 

274.1 

282.6 

288.5 

300.4 

7.8 

240.9 

245.9 

251.0 

256.3 

275.3 

284.0 

292.7 

298.9 

311.5 

7.9 

249.3 

254.5 

259.8 

265.2 

285.1 

294.1 

303.4 

309.7 

322.7 

8.0 

257.8 

263.2 

268.7 

274.4 

295.1 

304.2 

314.1 

321.1 

334.3 

8.1 

266.5 

272.1 

277.9 

283.7 

305.3 

315.0 

325.1 

331.9 

346.1 

8.2 

275.3 

281.2 

287.2 

290.8 

315.8 

325.9 

330.6 

343.4 

358.3 

8.3 

284.4 

290.4 

296.7 

303.1 

326.3 

336.9 

347.7 

355.2 

370.5 

8.4 

293.6 

300.0 

306.4 

313.0 

337.2 

348.2 

359.5 

367.2 

383.1 

8.5 

303.0 

309.6 

316.3 

323.2 

348.3 

359.6 

371.4 

379.4 

396.0 

8.6 

312.7 

319.5 

326.4 

333.5 

359.6 

371.4 

383.6 

391.1 

409.2 

8.7 

322.5 

329.6 

336.8 

344.1 

371.2 

383.4 

396.0 

404.7 

422.6 

8.8 

332.5 

339.8 

347.3 

354.9 

383.0 

395.7 

408.8 

417.8 

436.4 

8.9 

341.9 

349.4 

358.0 

365.9 

395.0 

408.2 

421.7 

431.1 

450.3 

9.0 

353. 

360.9 

368.9 

377.1 

407.2 

420.9 

435.0 

444.7 

464.6 

9.1 

363.6 

371.7 

380.1 

388.5 

419.7 

433.9 

448.4 

458.5 

479.2 

9.2 

374.1 

382.8 

382.4 

400.2 

432.5 

447.1 

462.2 

492.6 

494.1 

9.3 

385.3 

394.0 

402.9 

412.0 

445.5 

460.6 

476.3 

487.0 

509.3 

9.4 

396.5 

405.5 

414.7 

424.1 

458.7 

474.4 

490.6 

501.7 

524.7 

9.5 

407.9 

417.2 

426.6 

436.4 

472.1 

488.4 

505.2 

516.6 

540.5 

9.6 

419.5 

429.3 

438.9 

448.9 

485.9 

502.6 

520.0 

531.9 

556.5 

154 


THE  FLOW  OF  WATER 


TABLE  F.— Concluded. 


d  in 
Feet. 

m== 
0.95 

TO  = 

0.83 

m= 
0.70 

m= 
0.57 

m  = 
0.30 

m  = 
0.0 

K  = 
1.25 

K  = 
1.5 

K  = 

2.0 

d2'87 

d2.68 

d2'69 

d2'70 

d2.735 

d2-78 

d2'765 

d2'775 

d2'795 

9.7 

431.2 

441.1 

451.3 

461.6 

499.9 

517.2 

535.1 

547.4 

572.8 

9.8 

443.4 

453.6 

464.0 

474.6 

514.1 

531.9 

550.5 

563.3 

589.5 

9.9 

455.4 

466.0 

476.8 

487.8 

528.6 

547.1 

566.5 

579.3 

606.6 

10.0 

467.8 

478.7 

489.8 

501.2 

543.3 

562.4 

582.2 

595.7 

623.8 

10.5 

532.8 

545.5 

558.4 

571.8 

620.8 

658.1 

666.6 

682.0 

714.9 

11.0 

603.3 

617.9 

632.9 

648.4 

705.1 

730.9 

757.6 

776.0 

814.1 

11.5 

679.3 

696.1 

713.3 

730.9 

778.1 

827.8 

856.9 

878.0 

921.8 

12.0 

761.1 

780.2 

799.7 

820.1 

894.5 

928.4 

963.7 

987.9 

1038.3 

12.5 

848.7 

870.4 

892.6 

915.5 

1000.1 

1038.7 

1028.8 

1106.4 

1163.8 

13.0 

942.4 

966.9 

992.0 

1017.8 

1110.4 

1157.0 

1202.3 

1233.7 

1299.0 

13.5 

1042.3 

1069.8 

1098.0 

1135.0 

1234.4 

1284.0 

1335.0 

1370.0 

1443.0 

14.0 

1148.2 

1179.3 

1211.0 

1243.0 

1363.0 

1419.0 

1476.0 

1516.0 

1597.0 

14.5 

1261.0 

1296.0 

1330.0 

1399.0 

1501.0 

1562.0 

1626.0 

1670.0 

1762.0 

15.0 

1381.0 

1419.0 

1458.0 

1498.0 

1647.0 

1715.0 

1786.0 

1835.0 

1937.0 

15.5 

1509.0 

1549.0 

1592.0 

1637.0 

1801.0 

1877.0 

1956.0 

2010.0 

2125.0 

16.0 

1641.0 

1687.0 

1734.0 

1783.0 

1965.0 

2048  .  0 

2135.0 

2195.0 

2320.0 

16.5 

1781.0 

1832.0 

1884.0 

1937.0 

2137.0 

2229.0 

2325.0 

2391.0 

2528.0 

17.0 

1929.0 

1971.0 

2041.0 

2100.0 

2319.0 

2420.0 

2525.0 

2597.0 

2749.0 

17.5 

2084.0 

2145.0 

2207.0 

2271.0 

2569.0 

2620.0 

2135.0 

2815.0 

2980.0 

18.0 

2247.0 

2313.0 

2381.0 

2450.0 

2711.0 

2832.0 

2959.0 

3044.0 

3225.0 

18.5 

2417.0 

2489  .  0 

2563.0 

2639.0 

2922.0 

3053.0 

3264.0 

3284.0 

3481.0 

19.0 

2596.0 

2673.0 

2753.0 

2836.0 

3143.0 

3285.0 

3434.0 

3536.0 

3751.0 

19.5 

2782.0 

2866.0 

2952.0 

3042.0 

3375.0 

3528.0 

3689.0 

3801.0 

4033.0 

20.0 

2977.0 

3067.0 

3160.0 

3257.0 

3617.0 

3871.0 

3957.0 

4079.0 

4329.0 

21.0 

3391.0 

3496.0 

3604.0 

3715.0 

4133.0 

4329.0 

4528.0 

4668.0 

4962.0 

22.0 

3839.0 

3960.0 

4085.0 

4213.0 

4694.0 

4917.0 

5150.0 

5312.0 

5650.0 

23.0 

4323.0 

4461.0 

4603.0 

4750.0 

5300.0 

5556.0 

5823.0 

6009.0 

6395.0 

24.0 

4844.0 

5000.0 

5161.0 

5328.0 

5955.0 

6245.0 

6549.0 

6762.0 

7206.0 

25.0 

5401.0 

5578.0 

5761.0 

5949.0 

6658.0 

6988.0 

7334.0 

7573.0 

8077.0 

OPEN   CONDUITS 


155 


TABLE  G. 

QUANTITIES  OF  DISCHARGE  IN  CUBIC  FEET  PER  SECOND  OF  A 
CONDUIT  ONE  FOOT  IN  DIAMETER. 


Sec- 

Section Circular. 

tion 

Egg- 

Section  Circular. 

shaped 

Sine  of 

a  =  F* 

a  =  FT* 

a  =  1.0 

the  Slope. 

m  = 

m  = 

m= 

m  = 

m= 

TO  = 

m= 

m  = 

0.95 

0.83 

0.68 

0.57 

0.57 

0.53 

0.45 

0.30 

.000025 

0.1826 

0.1675 

0.1501 

0.2479 

0.1501 

0.1460 

0.1460 

0.1305 

.000030 

0.2023 

0.1856 

0.1663 

0.2730 

0.1663 

0.1608 

0.1608 

0.1430 

.000035 

0.2207 

0.2024 

0.1813 

0.2962 

0.1804 

0.1744 

0.1744 

0.1544 

.000040 

0.2378 

0.2182 

0.1955 

0.3179 

0.1941 

0.1872 

0.1872 

0.1651 

.000045 

0.2541 

0.2331 

0.2088 

0.3383 

0.2061 

0.1993 

0.1993 

0.1751 

.000050 

0.2697 

0.2473 

0.2216 

0.3577 

0.2179 

0.2107 

0.2107 

0.1846 

.000055 

0.2845 

0.2609 

0.2338 

0.3763 

0.2292 

0.2216 

0.2216 

0.1936 

.000060 

0.2988 

0.2740 

0.2455 

0.3940 

0.2400 

0.2321 

0.2321 

0.2021 

.000065 

0.3126 

0.2867 

0.2582 

0.4110 

0.2504 

0.2421 

0.2416 

0.2104 

.000070 

0.3259 

0.2989 

0.2678 

0.4275 

0.2604 

0.2518 

0.2507 

0.2183 

.000075 

0.3387 

0.3107 

0.2784 

0.4434 

0.2701 

0.2612 

0.2656 

0.2260 

.000080 

0.3513 

0.3222 

0.2887 

0.4588 

0.2725 

0.2702 

0.2681 

0.2334 

.000085 

0.3635 

0.3333 

0.2987 

0.4738 

0.2886 

0.2790 

0.2763 

0.2406 

.000090 

0.3753 

0.3442 

0.3084 

0.4884 

0.2975 

0.2876 

0.2843 

0.2496 

.000095 

0.3869 

0.3554 

0.3180 

0.5024 

0.3060 

0.2959 

0.2855 

0.2544 

.0001 

0.3983 

0.3652 

0.3349 

0.5164 

0.3145 

0.3071 

0.2997 

0.2610 

.000125 

0.4516 

0.4141 

0.3711 

0.5803 

0.3535 

0.3418 

0.3351 

0.2851 

.00015 

0.5003 

0.4588 

0.4111 

0.6400 

0.3898 

0.3769 

0.3671 

0.3271 

.000175 

0.5456 

0.8004 

0.4484 

0.6944 

0.4133 

0.4090 

0.3965 

0.3452 

.0002 

0.5881 

0.5408 

0.4833 

0.7453 

0.4539 

0.4390 

0.4239 

0.3691 

.000225 

0.6284 

0.5749 

0.5165 

0.7932 

0.4831 

0.4672 

0.4444 

0.4006 

.00025 

0.6668 

0.6258 

0.5480 

0.8387 

0.5108 

0.4940 

0.4739 

0.4127 

.000275 

0.7035 

0.6453 

0.5782 

0.8821 

0.5373 

0.5196 

0.4970 

0.4328 

.0003 

0.7388 

0.6776 

0.6071 

0.9225 

0.5629 

0.5433 

0.5191 

0.4520 

.00035 

0.8057 

0.7390 

0.6630 

1.0023 

0.6104 

0.5903 

0.5607 

0.4663 

.00040 

0.8686 

0.7785 

0.6817 

1.0759 

0.6553 

0.6337 

0.5994 

0.5220 

.00045 

0.9281 

0.8512 

0.7627 

1  .  1449 

0.6973 

0.6743 

0.6358 

0.5536 

.00050 

0.9848 

0.9030 

0.8092 

1.2105 

0.7373 

0.7130 

0.6701 

0.5836 

.00055 

1.0390 

0.9529 

0.8538 

1.273 

0.7755 

0.7329 

0.7048 

0.6138 

.00060 

1.0911 

.0007 

0.8967 

1.333 

0.8120 

0.7853 

0.7341 

0.6393 

.00065 

1.1414 

.0468 

0.9369 

1.391 

0.8475 

0.8195 

0.7641 

0.6654 

.00070 

1.1899 

.0913 

0.9779 

1.447 

0.8610 

0.8520 

0.7930 

0.6905 

.00075 

1.2370 

.1345 

1.0166 

1.501 

0.9139 

0.8837 

0.8208 

0.7149 

.00080 

1.283 

.1765 

.0541 

1.553 

0.9456 

0.9144 

0.8477 

0.7382 

.00085 

1.327 

.2173 

.0907 

1.603 

0.9765 

0.9447 

0.8738 

0.7609 

.00090 

1.371 

.257 

.1248 

1.653 

1.0065 

0.9733 

0.8991 

0.7830 

.00095 

1.413 

.296 

.1611 

1.70 

1.0357 

1.0015 

0.9238 

0.8044 

.001 

1.454 

1.334 

.1951 

1.747 

1.0642 

1.0291 

0.9477 

0.8253 

.0011 

1.535 

1.407 

.261 

1.838 

1.1193 

1.0823 

0.9940 

0.8656 

.0012 

1.611 

1.478 

.324 

1.924 

1.1721 

1.1334 

1.0382 

0.9041 

.0013 

1.686 

1.546 

1.385 

2.007 

1.2228 

1.1824 

1.0806 

0.9410 

156 


THE   FLOW   OF   WATER 


TABLE  G.  —  Continued. 


Sec- 

Section Circular. 

tion 

Egg- 

Section  Circular. 

shaped 

Sine  of 

a  =  v'* 

a  =  y" 

a=1.0 

the  Slope. 

m= 

m= 

TO  = 

m  = 

m= 

m  = 

m  = 

m  = 

0.95 

0.83 

0.68 

0.57 

0.57 

0.53 

0.45 

0.30 

.0014 

1.757 

1.612 

1.444 

2.088 

1.272 

1.2298 

1.1214 

0.9775 

.0015 

1.826 

1.676 

1.501 

2.165 

1.319 

1.276 

1.1608 

1.0108 

.0016 

1.894 

1.742 

1.557 

2.241 

1.365 

1.320 

1.1988 

1.0442 

.0017 

1.960 

1.798 

1.611 

2.314 

1.409 

1.363 

1.2357 

.0761 

.0018 

2.034 

1.856 

1.664 

2.385 

1.453 

1.405 

1.272 

.1083 

.0019 

2.089 

1.914 

1.715 

2.454 

.495 

1.446 

1.306 

.1386 

.0020 

2.148 

1.970 

1.765 

2.523 

.537 

1.486 

.340 

.1672 

.0021 

2.208 

2.025 

1.814 

2.588 

.576 

1.524 

.373 

.1960 

.0022 

2.270 

2.082 

1.865 

2.653 

.616 

1.562 

.406 

.2242 

.0023 

2.323 

2.131 

1.909 

2.715 

.654 

1.599 

.437 

.252 

.0024 

2.385 

2.183 

1.955 

2.777 

.692 

1.636 

.468 

1.279 

.0025 

2.435 

2.234 

2.001 

2.838 

.729 

1.672 

.498 

1.305 

.0026 

2.489 

2.283 

2.046 

2.898 

1.765 

1.707 

.528 

1.331 

.0027 

2.543 

2.332 

2.085 

2.956 

1.801 

1.741 

.557 

1.356 

.0028 

2.595 

2.381 

2.133 

3.014 

1.835 

1.775 

1.586 

1.381 

.0029 

2.647 

2.428 

2.175 

3.070 

1.870 

1.808 

1.614 

1.405 

.0030 

2.698 

2.474 

2.217 

3.126 

1.904 

1.841 

1.642 

1.430 

.0031 

2.748 

2.520 

2.259 

3.180 

1.937 

1.873 

1.669 

1.453 

.0032 

2.798 

2.566 

2.299 

3.234 

1.970 

1.905 

.695 

1.476 

.0033 

2.847 

2.610 

2.339 

3.288 

2.002 

1.936 

.722 

1.500 

.0034 

2.895 

2.655 

2.379 

3.340 

2.034 

1.967 

.748 

1.522 

.0035 

2.942 

2.699 

2.418 

3.391 

2.066 

1.996 

.773 

1.544 

.0036 

2.989 

2.742 

2.457 

3.442 

2.097 

2.028 

.798 

1.566 

.0037 

3.036 

2.784 

2.495 

3.493 

2.127 

2.057 

.823 

1.588 

.0038 

3.082 

2.826 

2.532 

3.542 

2.158 

2.086 

.848 

1.608 

.0039 

3.126 

2.867 

2.569 

3.591 

2.187 

2.115 

.872 

1.628 

.0040 

3.172 

2.909 

2.606 

3.640 

2.217 

2.144 

1.896 

1.651 

.0041 

3.216 

2.950 

2.643 

3.688 

2.246 

2.172 

1.919 

1.671 

.0042 

3.260 

2.990 

2.679 

3.735 

2.275 

2.200 

1.943 

1.691 

.0043 

3.304 

3.030 

2.715 

3.782 

2.304 

2.227 

1.965 

1.711 

.0044 

3.346 

3.069 

2.750 

3.829 

2.332 

2.255 

1.988 

1.731 

.0045 

3.389 

3.109 

2.786 

3.875 

2.360 

2.282 

2.011 

1.751 

.0046 

3.431 

3.147 

2.820 

3.920 

2.387 

2.308 

2.033 

1.770 

.0047 

3.473 

3.185 

2.854 

3.964 

2.415 

2.335 

2.055 

1.789 

.0048 

3.514 

3.223 

2.888 

4.009 

2.442 

2.361 

2.076 

1.808 

.0049 

3.555 

3.261 

2.922 

4.053 

2.468 

2.387 

2.098 

1.827 

.0050 

3.596 

3.298 

2.955 

4.096 

2.495 

2.413 

2.119 

1.845 

.0051 

3.636 

3.335 

2.988 

4.140 

2.521 

2.438 

2.140 

1.864 

.0052 

3.676 

3.372 

3.021 

4.182 

2.547 

2.463 

2.161 

1.882 

.0053 

3.716 

3.408 

3.053 

4.225 

2.573 

2.488 

2.182 

1.900 

.0054 

3.755 

3.444 

3.086 

4.265 

2.598 

2.517 

2.202 

1.917 

.0055 

3.794 

3.480 

3.118 

4.308 

2.623 

2.538 

2.223 

1.936 

.0056 

3.833 

3.515 

3.150 

4.350 

2.649 

2.562 

2.243 

1.953 

.0057 

3.871 

3.550 

3.181 

4.391  '2.674 

2.586 

2.262 

1.970 

OPEN   CONDUITS 


157 


TABLE   G.  —  Continued. 


Sec- 

Section Circular. 

tion 

Egg- 

Section  Circular. 

shaped 

Sine  of 

. 

. 

the  Slope. 

a==  V  9 

a=  FTS" 

a=1.0 

m  = 

m  — 







m  = 

m  — 

m= 

0.95 

0.83 

0.68 

0.57 

0.57 

0.53 

0.45 

0.30 

.0058 

3.909 

3.585 

3.212 

4.431 

2.699 

2.610 

2.282 

1.987 

.0059 

3.947 

3.620 

3.244 

4.471 

2.724 

2.634 

2.302 

2.005 

.0060 

3.984 

3.655 

3.274 

4.511 

2.748 

2.658 

2.322 

2.022 

.0061 

4.022 

3.689 

3.305 

4.551 

2.772 

2.681 

2.341 

2.038 

.0062 

4.060 

3.723 

3.335 

4.590 

2.796 

2.704 

2.360 

2.055 

.0063 

4.095 

3.756 

3.365 

4.630 

2.820 

2.727 

2.379 

2.072 

.0064 

4.132 

3.789 

3.395 

4.668 

2.843 

2.750 

2.398 

2.088 

.0065 

4.168 

3.823 

3.425 

4.707 

2.867 

2.772 

2.417 

2.104 

.0066 

4.204 

3.856 

3.455 

4.745 

2.890 

2.795 

2.435 

2.120 

.0067 

4.240 

3.887 

3.484 

4.783 

2.913 

2.817 

2.453 

2.136 

.0068 

4.275 

3.921 

3.513 

4.820 

2.936 

2.839 

2.472 

2.152 

.0069 

4.311 

3.953 

3.543 

4.858 

2.959 

2.861 

2.490 

2.168 

.0070 

4.345 

3.985 

3.571 

4.895 

2.982 

2.883 

2.508 

2.184 

.0071 

4.380 

4.017 

3.600 

4.932 

3.004 

2.905 

2.525 

2.199 

.0072 

4.415 

4.049 

3.629 

4.969 

3.026 

.926 

2.543 

2.215 

.0073 

4.449 

4.081 

3.656 

5.005 

3.049 

.948 

2.561 

2.230 

.0074 

4.483 

4.112 

3.684 

5.041 

3.071 

.969 

2.580 

2.245 

.0075 

4.517 

4.143 

3.712 

5.077 

3.093 

.990 

2.596 

2.261 

.0076 

4.551 

4.174 

3.740 

5.113 

3.114 

3.011 

2.613 

2.275 

.0077 

4.588 

4.208 

3.770 

5.148 

3.136 

3.032 

2.630 

2.290 

.0078 

4.619 

4.235 

3.795 

5.184 

3.157 

3.057 

2.647 

2.305 

.0079 

4.651 

4.266 

3.822 

5.219 

3.179 

3.074 

2.664 

2.320 

.0080 

4.684 

4.296 

3.849 

5.254 

3.200 

3.094 

2.681 

2.334 

.00825 

4.767 

4.372 

3.917 

5.340 

3.254 

3.145 

2.722 

2.370 

.0085 

4.847 

4.445 

3.983 

5.425 

3.304 

3.195 

2.763 

2.406 

.00875 

4.927 

4.518 

4.048 

5.509 

3.355 

3.244 

2.804 

2.441 

.0090 

5.005 

4.591 

4.113 

5.592 

3.406 

3.293 

2.843 

2.476 

.00925 

5.083 

4.662 

4.177 

5.673 

3.456 

3.342 

2.882 

2.510 

.0095 

5.160 

4.732 

4.240 

5.254 

3.505 

3.389 

2.921 

2.544 

.00975 

5.236 

4.802 

4.304 

5.834 

3.553 

3.436 

2.960 

2.577 

.010 

5.311 

4.871 

4.366 

5.913 

3.601 

3.482 

2.997 

2.610 

.01025 

5.885 

4.939 

4.425 

5.990 

3.649 

3.528 

3.032 

2.640 

.0105 

5.458 

5.006 

4.486 

6.067 

3.696 

3.573 

3.071 

2.674 

.01075 

5.531 

5.073 

4.545 

6.143 

3.742 

3.618 

3.107 

2.706 

.011 

5.603 

5.139 

4.605 

6.218 

3.788 

3.663 

3.143 

2.737 

.01125 

5.675 

5.204 

4.663 

6.293 

3.833 

3.706 

3.179 

2.968 

.01150 

5.745 

5.269 

4.721 

6.367 

3.878 

3.750 

3.214 

2.799 

.01175 

5.815 

5.333 

4.779 

6.439 

3.746 

3.793 

3.249 

2.829 

.012 

5.886 

5.397 

4.836 

6.511 

3.966 

3.835 

3.283 

2.859 

.01225 

5.953 

5.460 

4.892 

6.583 

4.010 

3.877 

3.317 

2.888 

.0125 

6.021 

5.523 

4.949 

6.653 

4.053 

3.919 

3.351 

2.918 

.01275 

6.089 

5.584 

5.004 

6.726 

4.096 

3.960 

3.384 

2.947 

.0130 

6.155 

5.645 

5.059 

6.793 

4.138 

4.001 

3.330 

2.967 

.01325 

6.222 

5.706 

5.113 

6.862 

4.180 

4.042 

3.450 

3.004 

158 


THE   FLOW   OF   WATER 
TABLE  G.  —  Concluded. 


Sec- 

Section Circular. 

tion 
Egg- 

Section  Circular. 

shaped 

Sine  of 

a=yi 

a=  FT* 

a=1.0 

the  Slope. 

m  = 

m  = 

m  = 

TO  = 

m  = 

m  = 

m  = 

771  = 

0.95 

0.83 

0.68 

0.57 

0.57 

0.53 

0.45 

).30 

.0135 

6.287 

5.767 

5.167 

6.931 

4.221 

4.082 

.482 

.032 

.01375 

6.353 

5.826 

5.220 

6.998 

4.263 

4.122 

.514 

.060 

.014 

6.417 

5.886 

5.274 

7.065 

4.303 

4.160 

.546 

.088 

.01425 

6.482 

5.945 

5.326 

7.132 

4.344 

4.201 

.578 

.116 

.0145 

6.545 

6.003 

5.379 

7.198 

4.384 

4.240 

.609 

.143 

.01475 

6.609 

6.062 

5.431 

7.263 

4.424 

4.278 

3.640 

3.170 

.015 

6.671 

6.119 

5.482 

7.328 

4.464 

4.316 

3.671 

3.196 

.01525 

6.734 

6.176 

5.534 

7.392 

4.503 

4.354 

3.700 

3.222 

.0155 

6.795 

6.238 

5.584 

7.456 

4.542 

4.392 

3.731 

3.249 

.01575 

6.857 

6.289 

5.635 

7.520 

4.580 

4.429 

3.761 

3.275 

.016 

6.917 

6.344 

5.684 

7.582 

4.618 

4.466 

3.791 

3.302 

.01625 

6.978 

6.400 

5.735 

7.645 

4.657 

4.503 

3.821 

3.327 

.0165 

7.038 

6.456 

5.784 

7.707 

4.695 

4.540 

3.850 

3.352 

.01675 

7.099 

6.510 

5.833 

7.769 

4.732 

4.576 

3.880 

3.379 

.0170 

7.158 

6.565 

5.882 

7.830 

4.769 

4.612 

3.909 

3.404 

.01725 

7.217 

6.619 

5.931 

7.891 

4.806 

4.648 

3.936 

3.428 

.0175 

7.276 

6.673 

5.979 

7.951 

4.843 

4.683 

3.965 

3.453 

.01775 

7.334 

6.726 

6.027 

8.011 

4.879 

4.718 

3.993 

3.477 

.018 

7.392 

6.779 

6.074 

8.071 

4.916 

4.754 

4.021 

3.501 

.01825 

7.450 

6.832 

6.122 

8.130 

4.952 

4.788 

4.049 

3.526 

.0185 

7.507 

6.885 

6.169 

8.189 

4.987 

4.823 

4.076 

3.549 

.01875 

7.564 

6.937 

6.215 

8.247 

5.023 

4.858 

4.104 

3.574 

.019 

7.620 

6.989 

6.262 

8.305 

5.059 

4.892 

4.131 

3.598 

.01925 

7.676 

7.040 

6.309 

8.363 

5.094 

4.926 

4.158 

3.621 

.0195 

7.732 

7.092 

6.354 

8.420 

5.128 

4.959 

4.185 

3.644 

.01975 

7.788 

7.142 

6.40 

8.477 

5.163 

4.993 

4.212 

3.668 

.02 

7.843 

7.193 

6.445 

8.534 

5.198 

5.026 

4.239 

3.691 

.0205 

7.952 

7.293 

6.535 

8.645 

5.266 

5.092 

4.291 

3.737 

.021 

8.061 

7.394 

6.625 

8.757 

5.334 

5.158 

4.343 

3.782 

.0215 

8.169 

7.492 

6.713 

8.867 

5.401 

5.222 

4.395 

3.827 

.022 

8.275 

7.590 

6.801 

8.977 

5.469 

5.286 

4.445 

3.871 

.0225 

8.389 

7.692 

6.892 

9.089 

5.536 

5.353 

4.496 

3.915 

.023 

8.485 

7.782 

6.972 

9.189 

5.597 

5.412 

4.545 

3.958 

.0235 

8.582 

7.871 

7.052 

9.294 

5.661 

5.474 

4.594 

4.001 

.024 

8.690 

7.970 

7.141 

9.398 

5.724 

5.536 

4.643 

4.043 

.0245 

8.792 

8.063 

7.225 

9.501 

5.787 

5.596 

4.691 

4.085 

.025 

8.893 

8.155 

7.307 

9.604 

5.850 

5.657 

4.739 

4.127 

.030 

9.852 

9.036 

8.09C 

10.577 

6.442 

6.230 

5.191 

4.485 

.040 

11.583 

10.623 

9.518 

12.  31" 

7.502 

7.255 

5.994 

5.221 

.050 

13.13 

12.052 

10.79 

13.86 

8.443 

8.164 

6.701 

5.836 

.060 

14.55 

13.34 

11.95' 

15.27 

9.298 

8.992 

7.341 

6.393 

.070 

15.90 

14.55 

13.40 

16.57 

10.089 

9.756 

7,929 

6.904 

.080 

17.11 

15.69 

14.05 

19.98 

10.828 

10.471 

8.477 

7.382 

.090 

18.28 

16.76 

15.02 

18.92 

11.525 

11.144 

8.992 

7.831 

0.10 

19.39 

17.79 

15.93 

20.0 

12.18 

11.784 

9.478 

8.253 

OPEN   CONDUITS 


159 


TABLE   H. 

VELOCITIES  OF  FLOW  IN  A  SEMISQUARE  ONE  FOOT  IN  DEPTH. 


Sine  of 
the 
Slope. 

o=  FT* 

a=1.0 

TO  = 
0.95 

m  = 
0.80 

rra  = 
0.70 

m  = 
0.57 

TO  — 
0.30 

TO  = 

0.0 

K  = 

1.2 

K  = 
1.5 

K  = 
1.93 

.000025 

0.397 

0.362 

0.3386 

0.3084 

0.2479 

0.1962 

0.1770 

0.1543 

0.1305 

.000030 

0.4372 

0.3986 

0.3730 

0.3396 

0.2730 

0.2150 

0.1939 

0.1691 

0.1430 

.000035 

0.4744 

0.4326 

0.4046 

0.3685 

0.2962 

0.2322 

0.2095 

0.1826 

0.  1544 

.000040 

0.5092 

0.4642 

0.4342 

0.3955 

0.3179 

0.2482 

0.2239 

0.  1952 

0.  1651 

.000045 

0.5419 

0.4941 

0.4621 

0.4192 

0.3383 

0.2642 

0.2375 

0.2070 

0.1751 

.000050 

0.5863 

0.5346 

0.5001 

0.4451 

0.3577 

0.2775 

0.2503 

0.2182 

0.1846 

.000055 

0.6027 

0.5495 

0.5139 

0.4681 

0.3763 

0.2911 

0.2625 

0.2289 

0.1936 

.000060 

0.6311 

0.5754 

0.5382 

0.4902 

0.3940 

0.3040 

0.2742 

0.2391 

0.2021 

.000065 

0.6584 

0.6003 

0.5615 

0.5114 

0.4110 

0.3164 

0.2854 

0.2488 

0.2104 

.000070 

0.6847 

0.6243 

0.5839 

0.5318 

0.4275 

0.3283 

0.2962 

0.2582 

0.2183 

.000075 

0.7102 

0.6475 

0.6057 

0.5516 

0.4434 

0.3399 

0.3066 

0.2673 

0  .  2260 

.000080 

0.7349 

0.6701 

0.6267 

0.5708 

0.4588 

0.3510 

0.3166 

0.2761 

0.2334 

.000085 

0.7588 

0.6919 

0.6472 

0.5894 

0.4738 

0.3618 

0.3264 

0.2845 

0  .  2406 

.000090 

0.7822 

0.7131 

0.6671 

0.6075 

0.4884 

0.3723 

0.3358 

0.2928 

0.2476 

.000095 

0.8048 

0.7338 

0.6863 

0.6251 

0.5024 

0.3825 

0.3450 

0.3008 

0.2544 

.0001 

0.8720 

0.7541 

0.7053 

0.6424 

0.5164 

0.3924 

0.3540 

0.3086 

0.2610 

.000125 

0.9295 

0.8475 

0.7927 

0.7220 

0.5803 

0.4388 

0.3958 

0.3450 

0.2851 

.00015 

1.025 

0.9346 

0.8742 

0.7961 

0.646 

0.4806 

0.4437 

0.3868 

0.3271 

.000175 

1.1122 

1.0141 

0.9485 

0.8638 

0.6944 

0.5191 

0.4683 

0.4083 

0.3452 

.0002 

1.1937 

1.0884 

1.0180 

0.9272 

0.7453 

0.555 

0.5006 

0.4365 

0.3691 

.000225 

1.270 

1.1584 

1.0835 

0.9869 

0.7932 

0.5885 

0.5433 

0.4737 

0  .  4006 

.00025 

1.3404 

1  .  2248 

1.1457 

1.0423 

0.8387 

0.6205 

0.5597 

0.4880 

0.4127 

.000275 

1.413 

1.289 

1.2044 

1.0975 

0.8821 

0.6508 

0.5870 

0.5118 

0  .  4328 

.0003 

1.477 

1.347 

1.260 

1.1476 

0.9225 

0.6797 

0.6131 

0.5345 

0  .  4520 

.00035 

1.605 

1.464 

1.369 

.2444 

1.0023 

0.7342 

0.6623 

0.5774 

0  .  4663 

.00040 

1.723 

1.571 

1.470 

.338 

1.0759 

0.7852 

0.7080 

0.6172 

0  .  5220 

.00045 

1.834 

1.672 

1.564 

.424 

1  .  1449 

0.8325 

0.7509 

0.6547 

0.5536 

.0005 

1.939 

1.768 

1.654 

.506 

.2125 

0.8775 

0.7915 

0.6901 

0  .  5836 

.00055 

2.039 

1.859 

1.739 

.584 

.273 

0.9229 

0.8325 

0.7258 

0.6138 

.0006 

2.135 

1.947 

1.821 

.659 

.333 

0.9613 

0.8671 

0.7559 

0  .  6393 

.00065 

2.229 

2.032 

1.901 

.731 

.391 

1.0005 

0.9025 

0.7868 

0  .  6654 

.0007 

2.317 

2.113 

1.976 

.800 

.447 

1.0383 

0.9366 

0.8165 

0  .  6905 

.00075 

2.403 

2.191 

2.050 

.867 

.501 

.0750 

0.9697 

0.8454 

0.7149 

.0008 

2.487 

2.269 

2.121 

.932 

.553 

.110 

1.0012 

0.8729 

0.7382 

.00085 

2.568 

2.341 

2.192 

.995 

.603 

.1441 

1.032 

0.8998 

0  .  7609 

.0009 

2.647 

2.413 

2.257 

2.056 

.653 

.1774 

1.062 

0.9259 

0  .  7830 

.00095 

2.724 

2.483 

2.323 

2.116 

.70 

.2096 

1.091 

0.9512 

0  .  8044 

.001 

2.799 

2.551 

2.388 

2.174 

.747 

.241 

1.1194 

0.9759 

0  .  8253 

.0011 

2.943 

2.684 

2.510 

2.286 

1.838 

.302 

1.174 

1.0236 

0.8656 

.0012 

3.082 

2.810 

2.629 

2.394 

1.924 

.360 

1.2262 

.0691 

0.9041 

.0013 

3.215 

2.932 

2.742 

2.498 

2.007 

.404 

1.276 

.1127 

0.9410 

.0014 

3.344 

3.049 

2.852 

2.598 

2.088 

.468 

1.325 

.1548 

0.9775 

.0015 

3.469 

3.163 

2.958 

2.694 

2.165 

.520 

1.371 

.1953 

1.0108 

.0016 

3.589 

3.273 

3.061 

2.788 

2.241 

.570 

1.416 

.2347 

1  .  0442 

.0017 

3.706 

3.379 

3.161 

2.879 

2.314 

.621 

1.460 

.267 

1.0761 

.0018 

3.820 

3.483 

3.258 

2.967 

2.385 

1.665 

1.502 

.309 

1.1083 

.0019 

3.931 

3.584 

3.353 

3.053 

2.454 

1.711  1.543 

.346 

1.1388 

160 


THE   FLOW   OF  WATER 


TABLE   H.  —  Continued. 


Sine  of 

a=FT* 

a= 

1.0 

the 
Slope. 

m= 
0.95 

7?J,= 

0.80 

m  = 
0.70 

m  = 
0.57 

m  = 
0.30 

m  = 
0.0 

K  = 

1.2 

K= 
1.5 

tf= 
1.93 

.0020 

4.041 

3.685 

3.446 

3.139 

2.523 

1.755 

1.583 

1.380 

.1672 

.0021 

4.145 

3.779 

3.535 

3.220 

2.588 

1.798 

.622 

.414 

.1960 

.0022 

4.248 

3.874 

3.623 

3.300 

2.653 

1.841 

.660 

.447 

.2242 

.0023 

4.349 

3.966 

3.709 

3.378 

2.715 

1.882 

.698 

.480 

.252 

.0024 

4.449 

4.056 

3.794 

3.456 

2.  777 

1.923 

.734 

.521 

.279 

.0025 

4.546 

4.145 

3.877 

3.532 

2.838 

1.962 

.770 

.543 

.305 

.0026 

4.641 

4.232 

3.958 

3.605 

2.898 

2.001 

.805 

.574 

.331 

.0027 

4.735 

4.317 

4.038 

3.678 

2.956 

2.039 

1.839 

.604 

.356 

.0028 

4.827 

4.401 

4.117 

3.749 

3.014 

2.076 

1.873 

.633 

.381 

.0029 

4.918 

4.484 

4.194 

3.820 

3.070 

2.114 

1.906 

.662 

.405 

.0030 

5.008 

4.566 

4.271 

3.890 

3.126 

2.149 

1.939 

.690 

.430 

.0031 

5.094 

4.645 

4.345 

3.957 

3.180 

2.185 

1.971 

.718 

1.453 

.0032 

5.180 

4.724 

4.418 

4.024 

3.234 

2.220 

2.003 

.746 

1.476 

.0033 

5.265 

4.801 

4.491 

4.090 

3.288 

2.254 

2.034 

.773 

1.500 

.0034 

5.350 

4.898 

4.562 

4.155 

3.340 

2.288 

2.064 

.800 

1.522 

.0035 

5.432 

4.953 

4.633 

4.219 

3.391 

2.322 

2.094 

.826 

1.544 

.0036 

5.514 

5.027 

4.703 

4.283 

3.442 

2.355 

2.124 

.852 

1.566 

.0037 

5.594 

5.101 

4.771 

4.346 

3.493 

2.389 

2.153 

.877 

.588 

.0038 

5.674 

5.173 

4.839 

4.407 

3.542 

2.419 

2.182 

1.902 

.608 

.0039 

5.753 

5.245 

4.906 

4.468 

3.591 

2.451 

2.211 

1.928 

.628 

.0040 

5.830 

5.316 

4.972 

4.528 

3.640 

2.482 

2.239 

1.952 

.651 

.0041 

5.907 

5.386 

5.037 

4.588 

3.688 

2.513 

2.267 

1.976 

.671 

.0042 

5.982 

5.455 

5.102 

4.647 

3.735 

2.543 

2.294 

2.000 

.691 

.0043 

6.057 

5.523 

5.166 

4.705 

3.782 

2.573 

2.321 

2.024 

1.711 

.0044 

6.132 

5.591 

5.229 

4.763 

3.829 

2.603 

2.348 

2.047 

1.731 

.0045 

6.206 

5.659 

5.293 

4.820 

3.875 

2.633 

2.375 

2.070 

1.751 

.0046 

6.278 

5.724 

5.354 

4.876 

3.920 

2.662 

2.401 

2.093 

1.770 

.0047 

6.350 

5.790 

5.415 

4.932 

3.964 

2.691 

2.427 

2.116 

1.789 

.0048 

6.421 

5.854 

5.476 

4.987 

4.009 

2.719 

2.452 

2.138 

1.808 

.0049 

6.491 

5.919 

5.536 

5.042 

4.053 

2.747 

2.448 

2.160 

1.827 

.0050 

6.561 

5.982 

5.595 

5.096 

4.096 

2.775 

2.503 

2.182 

1.845 

.0051 

6.630 

6.045 

5.654 

5.150 

4.140 

2.802 

2.528 

2.204 

1.864 

.0052 

6.698 

6.108 

5.713 

5.203 

4.182 

2.830 

2.553 

2.226 

1.882 

.0053 

6.767 

6.170 

5.771 

5.256 

4.225 

2.857 

2.577 

2.247 

1.900 

.0054 

6.832 

6.230 

5.827 

5.307 

4.265 

2.884 

2.601 

2.268 

1.917 

.0055 

6.901 

6.292 

5.885 

5.360 

4.308 

2.910 

2.625 

2.289 

1.936 

.0056 

6.967 

6.352 

5.941 

5.412 

4.350 

2.937 

2.649 

2.310 

1.953 

.0057 

7.032 

6.412 

5.997 

5.462 

4.391 

2.962 

2.672 

2.330 

1.970 

.0058 

7.097 

6.471 

6.053 

5.513 

4.431 

2.989 

2.695 

2.350 

1.987 

.0059 

7.162 

6.530 

6.108 

5.563 

4.471 

3.014 

2.719 

2.370 

.005 

.0060 

7.226 

6.588 

6.162 

5.613 

4.511 

3.040 

2.742 

2.390 

.022 

.0061 

7.289 

6.646 

6.217 

5.662 

4.551 

3.065 

2.765 

2.410 

.038 

.0062 

7.352 

6.704 

6.270 

5.711 

4.590 

3.090 

2.787 

2.430 

.055 

.0063 

7.415 

6.761 

6.324 

5.759 

4.630 

3.115 

2.810 

2.450 

.072 

.0064 

7.477 

6.818 

6.377 

5.808 

4.668 

3.140 

2.832 

2.469 

.088 

.0065 

7.539 

6.873 

6.429 

5.856 

4.707 

3.164 

2.854 

2.488 

2.104 

.0066 

7.600 

6.930 

6.481 

5.903 

4.745 

3.188 

2.876 

2.507 

2.120 

.0067 

7.660 

6.985 

6.533 

5.951 

4.783 

3.212 

2.897 

2.526 

2.136 

OPEN  CONDUITS 


161 


TABLE  H.  —  Continued. 


Sine  of 
the 
Slope. 

a=  FT* 

a—  1.0 

m  = 
0.95 

m  = 
0.80 

m  = 
0.70 

m= 
0.57 

m  = 
0.30 

m= 
0.0 

K= 

1.2 

K  = 

1.5 

&'= 
1.93 

.0068 

7.721 

7.040 

6.585 

5.997 

4.820 

3.236 

2.919 

2.545 

2.152 

.0069 

7.781 

7.095 

6.636 

6.044 

4.858 

3.260 

2.940 

2.564 

2.168 

.0070 

7.840 

7.149 

6.686 

6.090 

4.895 

3.283 

2.962 

2.582)  2.184 

.0071 

7.899 

7.202 

6.737 

6.136 

4.932 

3  .  307 

2.983 

2.601 

2.199 

.0072 

7.958 

7.256 

6.787 

6.181 

4.969 

3.330 

3.004 

2.619 

2.215 

.0073 

8.017 

7:309 

6.837 

6.227 

5.005 

3.353 

3.024 

2.637 

2.230 

.0074 

8.075 

7.362 

6.886 

6.272 

5.041 

3.376 

3.045 

2.655 

2.245 

.0075 

8.132 

7.414 

6.935 

6.321 

5.077 

3.399 

3.067 

2.673 

2.261 

.0076 

8.189 

7.467 

6.984 

6.361 

5.113 

3.421 

3.086 

2.690 

2.275 

.0077 

8.246 

7.519 

7.033 

6.405 

5.148 

3.444 

3.106 

2.708 

2.29 

.0078 

8.303 

7.571 

7.081 

6.449 

5.184 

3.466 

3.126 

2.726 

2.305 

.0079 

8.359 

7.622 

7.129 

6.493 

5.219 

3.488 

3.146 

2.743 

2.320 

.0080 

8.415 

7.672 

7.176 

6.536 

5.254 

3.510 

3.166 

2.760 

2.334 

.00825 

8.553 

7.799 

7.294 

6.644 

5.340 

3.565 

3.215 

2.803 

2.370 

.0085 

8.689 

7.923 

7.410 

6.757 

5.425 

3.618 

3.264 

2.845 

2.406 

.00875 

8.824 

8.045 

7.525 

6.853 

5.509 

3.671 

3.311 

2.887 

2.411 

.0090 

8.957 

8.167 

7.639 

6.957 

5.592 

3.723 

3.358 

2.928 

2.476 

.00925 

9.087 

8.286 

7.749 

7.058 

5.673 

3.774 

3.405 

2.968 

2.510 

.0095 

9.216 

8.403 

7.860 

7.159 

5.754 

3.825 

3.450 

3.008 

2.544 

.00975 

9.344 

8.520 

7.969 

7.258 

5.834 

3.895 

3.495 

3.048 

2.577 

.01 

9.476 

8.635 

8.076 

7.356 

5.913 

3.924 

3.540 

3.086 

2.610 

.01025 

9.594 

8.748 

8.182 

7.453 

5.990 

3.970 

3.581 

3.122 

2.640 

.0105 

9.718 

8.860 

8.288 

7.548 

6.067 

4.021 

3.627 

3.163 

2.674 

.01075 

9.839 

8.971 

8.391 

7.643 

6.143 

4.069 

3.670 

3.200 

2.706 

.011 

9.960 

9.082 

8.494 

7.736 

6.218 

4.116 

3.713 

3.237 

2.737 

.01125 

10.079 

9.188 

8.596 

7.829 

6.293 

4.163 

3.755 

3.274 

2.768 

.0115 

10.197 

9.298 

8.696 

7.921 

6.367 

4.208 

3.796 

3.310 

2.799 

.01175 

10.314 

9.404 

8.796 

8.011 

6.439 

4.254 

3.837 

3.345 

2.829 

.012 

10.430 

9.510 

8.895 

8.101 

6.511 

4.289 

3.869 

3.381 

2.859 

.01225 

10.544 

9.614 

8.992 

8.190 

6.583 

4.343 

3.918 

3.415 

2.888 

.0125 

10.658 

9.717 

9.089 

8.278 

6.653 

4.388 

3.958 

3.450 

2.918 

.01275 

10.772 

9.822 

9.187 

8.367 

6.726 

4.431 

3.997 

3.485 

2.947 

.013 

10.881 

9.921 

9.280 

8.452 

6.793 

4.461 

4.024 

3.509 

2.967 

.01325 

10.991 

10.022 

9.374 

8.537 

6.862 

4.517 

4.074 

3.552 

3.004 

.0135 

11.10 

10.121 

9.467 

8.622 

6.931 

4.560 

4.113 

3.586 

3.032 

.01375 

11.209 

10.220 

9.559 

8.706 

6.998 

4.602 

4.151 

3.619 

3.060 

.014 

11.316 

10.318 

9.652 

8.790 

7.065 

4.643 

4.188 

3.652 

3.088 

.01425 

11.423 

10.415 

9.742 

8.873 

7.132 

4.685 

4.226 

3.684 

3.116 

.0145 

11.528 

10.511 

9.832 

8.954 

7.198 

4.726 

4.263 

3.716 

3.143 

.01475 

11.633 

10.607 

9.921 

9.036 

7.263 

4.766 

4.299 

3.748 

3.170 

.015 

11.737 

10.702 

10.01 

9.117 

7.328 

4.806 

4.336 

3.780 

3.196 

.01525 

11.840 

10.796 

10.098 

9.197 

7.392 

4.845 

4.370 

3.810 

3.222 

.0155 

11.943 

10.890 

10.185 

9.276 

7.456 

4.886 

4.407 

3.842 

3.249 

.01575 

12.025 

10.981 

10.272 

9.373 

7.520 

4.925 

4.442 

3.873 

3.275 

.016 

12.230 

11.073 

10.357 

9.433 

7.582 

4.965 

4.479 

3.905 

3.302 

.01625 

12.274 

11.165 

10.443 

9.512 

7.645 

5.003 

4.513 

3.934 

3.327 

.0165 

12.345 

11.256 

10.522 

9.589 

7.707 

5.041 

4.547 

3.964 

3.352 

.01675 

12.443 

11.346 

10.612 

9.665 

7.769 

5.080 

4.582 

3.995 

3.376 

.017 

12.54 

11.435 

10.696 

9.742 

7.830 

5.118 

4.617 

4.025 

3.404 

162 


THE   FLOW    OF   WATER 


TABLE   H. —  Concluded. 


Sine  of 
the 
Slope 

a=yT8 

a=1.0 

w= 
0.95 

m  = 
0.80 

w  = 
0.70 

m  = 
0.57 

m  = 
0.30 

7tt  = 

0.0 

K  = 
1.2 

K  = 
1.5 

tf= 
1.93 

.01725 

12.64 

11.524 

10.779 

9.817 

7.891 

5.154 

4.649 

4.053 

3.428 

.0175 

12.74 

11.612 

10.861 

9.892 

7.951 

5.191 

4.683 

4.083 

3.453 

.01775 

12.83 

11.700 

10.943 

9.967 

8.011 

5.228 

4.716 

4.112 

3.477 

.018 

12.92 

11.786 

11.024 

10.041 

8.091 

5.265 

4.749 

4.141 

3.501 

.01825 

13.02 

11.873 

11.105 

10.115 

8.130 

5.302 

4.782 

4.169 

3.526 

.0185 

13.11 

11.959 

11.185 

10.188 

8.189 

5.338 

4.815 

4.198 

3.549 

.01875 

13.21 

12.044 

11.265 

10.260 

8.247 

5.374 

4.847 

4.226 

3.574 

.019 

13.33 

12.157 

11.344 

10.332 

8.305 

5.409 

4.879 

4.254 

3.598 

.01925 

13.40 

12.213 

11.423 

10.404 

8.363 

5.445 

4.911 

4.282 

3.621 

.0195 

13.48 

12.297 

11.502 

10.475 

8.420 

5.480 

4.943 

4.310 

3.644 

.01975 

13.58 

12.380 

11.579 

10.546 

8.477 

5.515 

4.975 

4.337 

3.668 

.020 

13.67 

12.46 

11.657 

10.616 

8.534 

5.550 

5.006 

4.365 

3.691 

.0205 

13.85 

12.62 

11.809 

10.756 

8.645 

5.619 

5.668 

4.419 

3.737 

.021 

14.03 

12.79 

11.962 

10.895 

8.757 

5.687 

5.129 

4.472 

3.782 

.0215 

14.20 

12.95 

12.112 

11.031 

8.867 

5.754 

5.191 

4.525 

3.827 

,022 

14.38 

13.11 

12.263 

11.169 

8.977 

5.821 

5.251 

4.578 

3.871 

.0225 

14.56 

13.27 

12.415 

11.308 

9.089 

5.887 

5.310 

4.630 

3.915 

.023 

14.72 

13.42 

12.55 

11.432 

9.189 

5.952 

5.369 

4.680 

3.958 

.0235 

14.89 

13.57 

12.70 

11.563 

9.294 

6.016 

5.427 

4.731 

4.001 

.024 

15.05 

13.72 

12.84 

11.693 

9.398 

6.080 

5.484 

4.781 

4.043 

.0245 

15.22 

13.88 

12.98 

11.821 

9.501 

6.143 

5.541 

4.831 

4.085 

.025 

15.38 

14.03 

13.12 

11.948 

9.604 

6.205 

5.597 

4.880 

4.127 

.030 

16.94 

15.45 

14.45 

13.160 

10.577 

6.797 

6.131 

5.345 

4.485 

.040 

19.73 

17.99 

16.83 

15.32 

12.317 

7.851 

7.082 

6.174 

5.221 

,050 

22.20 

20.24 

18.93 

17.25 

13.860 

8.775 

7.915 

6.901 

5.836 

.060 

24.45 

22.30 

20.85 

18.99 

15.27 

9.613 

8.671 

7.559 

6.393 

.070 

26.53 

24.19 

22.63 

20.61 

16.57 

10.383 

9.366 

8.165 

6.904 

.080 

28.48 

25.96 

24.28 

22.12 

17.78 

11.10 

10.012 

8.729 

7.382 

.090 

30.31 

27.63 

26.45 

22.48 

18.92 

11.762 

10.621 

9.259 

7.831 

.100 

32.04 

29.22 

27.33 

24.89 

20.00 

12.41 

11.194 

9.760 

8.253 

OPEN   CONDUITS 


163 


TABLE  I. 

QUANTITIES  OP  DISCHARGE  IN  CUBIC  FEET  PER  SECOND  OP  A  SEMI- 
SQUARE  ONE  FOOT  IN  DEPTH. 


Sine  of 

a=FT 

k 

a= 

1.0 

the 
Slope 

m  = 
0.95 

m= 
0.80 

m  = 
0.70 

m  = 
0.57 

m  = 
0.30 

m  = 
0.0 

K  = 
1.2 

K  = 
1.5 

K  = 
1.93 

.000025 

0.794 

0.724 

0.6772 

0.6168 

0.4958 

0.3924 

0.3540 

0.3086 

0.2610 

.000030 

0.8744 

0.7972 

0.7560 

0.6792 

0.5460 

0.4300 

0.3878 

0.3382 

0.2860 

.000035 

0.9488 

0.8652 

0.8092 

0.7370 

0.5924 

0.4644 

0.4190 

0.3652 

0.3088 

.000040 

1.0184 

0.9284 

0.8684 

0.7910 

0.6358 

0.4964 

0.4478 

0.3904 

0.3302 

.000045 

1.0838 

0.9982 

0.9242 

0.8384 

0.6766 

0.5284 

0.4750 

0.4140 

0.3502 

.000050 

1.1726 

1.0692 

1.0002 

0.8902 

0.7154 

0.5550 

0.5006 

0.4364 

0.3692 

.000055 

1.2054 

1.0990 

1.0278 

0.9362 

0.7526 

0.5822 

0.5250 

0.4578 

0.3892 

.000060 

1.2622 

1.1508 

1.0764 

0.9804 

0.7880 

0.6080 

0.5484 

0.4782 

0.4042 

.000065 

1.2168 

1.2006 

1.1230 

.0228 

0.8220 

0.6328 

0.5708 

0.4976 

0.4208 

.000070 

1.3694 

1  .  2486 

1.1678 

.0636 

0.8550 

0.6566 

0.5924 

0.5164 

0.4366 

.000075 

1.4204 

1.3950 

1.2114 

.1032 

0.8868 

0.6798 

0.6132 

0.5346 

0.4520 

.000080 

1.4698 

1  .  3402 

1.2534 

.1416 

0.9176 

0.7020 

0.6332 

0.5522 

0.4668 

.000085 

1.5176 

1.3938 

1  .  2944 

.1788 

0.9476 

0.7236 

0.6528 

0.5690 

0.4812 

.000090 

1.5644 

1.4262 

1.3342 

.2150 

0.9768 

0.7446 

0.6716 

0.5856 

0.4952 

.000095 

1.6096 

.4676 

1.3726 

1.2502 

1.0048 

0.7650 

0.6900 

0.6016 

0.5088 

.0001 

1  .  6440 

.5082 

1.4106 

1.2848 

1.0328 

0.7848 

0.7080 

0.6172 

0.5220 

.000125 

1.8590 

.6950 

1.5854 

1.4440 

1.1606 

0.8776 

0.7916 

0.6900 

0.5702 

.00015 

2.050 

.8692 

1.7484 

1.5922 

1.280 

0.9612 

0.8874 

0.7736 

0.6542 

.000175 

2.2244 

.0282 

1.8970 

1.7276 

1.3888 

1.0382 

0.9366 

0.8166 

0.6904 

.0002 

2.3894 

.1768 

2.0360 

1.8544 

1  .  4906 

1.110 

1.0012 

0.8730 

0.7382 

.000225 

2.540 

.3168 

2.1670 

1.9738 

1.5864 

1.1770 

1.0866 

0.9474 

0.8002 

.00025 

2.6808 

.4496 

2.2914 

2.0866 

1.6774 

1.2410 

1.1194 

0.9760 

0.8254 

.000275 

2.826 

.578 

2.4088 

2.1950 

1  .  7642 

1.3016 

.1740 

1.0236 

0.8656 

.0003 

2.954 

2.694 

2.520 

2.2952 

1  .  8450 

1.3594 

.2262 

1.0690 

0.9040 

.00035 

3.210 

2.928 

2.738 

2.4888 

2.0046 

1.4684 

.3246 

1.1548 

0.9326 

.00040 

3.446 

3.142 

2.940 

2.676 

2.1518 

.5704 

.416 

1.2344 

.044 

.00045 

3.668 

3.344 

3.128 

2.848 

2.2898 

.6650 

.5018 

1.3094 

.1072 

.0005 

3.878 

3.536 

3  .  308 

3.012 

2.4250 

.7550 

.5830 

1.3802 

.1672 

.00055 

4.078 

3.718 

3.478 

3.168 

2.546 

.8458 

.6650 

1.4516 

.2276 

.0006 

4.270 

3.894 

3.642 

3.318 

2.666 

.9226 

1.7342 

1.5118 

.2786 

.00065 

4.458 

4.064 

3.802 

3.462 

2.782 

2.001 

1.8050 

1.5936 

.3308 

.0007 

4.634 

4.226 

3.952 

3.600 

2.894 

2.0766 

1.8732 

1.6330 

.3810 

.00075 

4.806 

4.382 

4.100 

3.734 

3.002 

2.150 

1.9394 

1.6908 

.4298 

.0008 

4.974 

4.534 

4.242 

3.864 

3.106 

2.220 

2.0024 

1.7458 

.4764 

.00085 

5.136 

4.682 

4.394 

3.990 

3.206 

2.2882 

2.064 

1.7996 

.5218 

.0009 

5.294 

4.826 

4.514 

4.112 

3.306 

2.3548 

2.124 

1.8518 

.5660 

.00095 

5.448 

4.966 

4.646 

4.232 

3.40 

2.4192 

2.182 

1.9024 

1.6088 

.001 

5.598 

5.102 

4.776 

4.348 

3.494 

2.482 

2.2398 

1.9518 

1.6506 

.0011 

5.886 

5.368 

5.020 

4.572 

3.676 

2.604 

2.348 

2.0472 

1.7312 

.0012 

6.164 

5.620 

5.258 

4.798 

3.848 

2.720 

2.4524 

2.1382 

1.8082 

.0013 

6.430 

5.864 

5.484 

4.996 

4.014 

2.808 

2.552 

2.2254 

1.882 

.0014 

6.688 

6.098 

5.704 

5.196 

4.176 

2.936 

2.650 

2.3096 

1.9550 

.0015 

6.938 

6.326 

5.916 

5.388 

4.330 

3.04 

2.742 

2.3906 

2.0216 

.0016 

7.178 

6.546 

6.122 

5.576 

4.482 

3.14 

2.832 

2.4694 

2.0884 

.0017 

7.412 

6.758 

6.322 

5.758 

4.628 

3.242 

2.92 

2.534 

2.1562 

164 


THE   FLOW   OF   WATER 


TABLE  I.  —  Continued. 


Sine  of 
the 
Slope 

a=FT¥ 

a=1.0 

m= 
0.95 

m= 
0.80 

m= 
0.70 

m= 
0.57 

m  = 
0.30 

TO  = 
0.0 

K  = 
1.2 

K= 

1.5 

K  = 
1.93 

.0018 

7.640 

6.966 

6.516 

5.934 

4.770 

3.330 

3.004 

2.618 

2.2166 

.0019 

7.862 

7.168 

6.706 

6.106 

4.908 

3.422 

3.086 

2.692 

2.2772 

.0020 

8.082 

7.370 

6.892 

6.278 

5.046 

3.510 

3.166 

2.760 

2.3344 

.0021 

8.290 

7.558 

7.070 

6.440 

5.176 

3.596 

3.244 

2.828 

2.3920 

.0022 

8.496 

7.948 

7.246 

6.600 

5.306 

3.682 

3.320 

2.894 

2.4484 

.0023 

8.698 

7.932 

7.418 

6.756 

5.430 

3.764 

3.396 

2.960 

2.504 

.0024 

8.898 

8.112 

7.588 

6.912 

5.554 

3.846 

3.468 

3.042 

2.558 

.0025 

9.092 

8.290 

7.754 

7.064 

5.676 

3.924 

3.54 

3.086 

2.610 

.0026 

9.282 

8.  -464 

7.916 

7.210 

5.796 

4.002 

3.610 

3.148 

2.662 

.0027 

9.470 

8.634 

8.076 

7.356 

5.912 

4.078 

3.678 

3.208 

2.712 

.0028 

9:654 

8.802 

8.234 

7.498 

6.028 

4.152 

3.746 

3.266 

2.762 

.0029 

9.836 

8.968 

8.388 

7.64 

6.14 

4.228 

3.812 

3.324 

2.810 

.0030 

10.016 

9.132 

8.542 

7.78 

6.252 

4.298 

3.878 

3.38 

2.86 

.0031 

10.188 

9.390 

8.690 

7.914 

6.36 

4.370 

3.942 

3.436 

2.906 

.0032 

10.360 

9.448 

8.836 

8.048 

6.464 

4.440 

4.006 

3.492 

2.952 

.0033 

10.530 

9.602 

8.982 

8.18 

6.576 

4.508 

4.068 

3.546 

3.0 

.0034 

10.70 

9.956 

9.124 

8.310 

6.68 

4.576 

4.128 

3.60 

3.044 

.0035 

10.864 

9.906 

9.266 

8.438 

6.782 

4.644 

4.188 

3.652 

3.088 

.0036 

11.028 

10.054 

9.406 

8.566 

6.884 

4.710 

4.248 

3.704 

3.132 

.0037 

11.188 

10.202 

9.542 

8.692 

6.986 

4.774 

4.306 

3.754 

3.176 

.0038 

11.348 

10.346 

9.678 

8.814 

7.084 

4.838 

4.364 

3.804 

3.216 

.0039 

11.506 

10.490 

9.812 

8.936 

7.182 

4.902 

4.422 

3.856 

3.256 

.0040 

11.66 

10.632 

9.944 

9.056 

7.28 

4.964 

4.478 

3.904 

3.302 

.0041 

11.814 

10.772 

10.074 

9.176 

7.376 

5.026 

4.434 

3.952 

3.342 

.0042 

11.964 

10.910 

10.204 

9.294 

7.570 

5.086 

4.588 

4.0 

3.382 

.0043 

12.114 

11.046 

10.332 

9.410 

7.564 

5.146 

4.642 

4.048 

3.422 

.0044 

12.264 

11.182 

10.458 

9.526 

7.658 

5.206 

4.696 

4.094 

3.462 

.0045 

12.412 

11.318 

10.586 

9.64 

7.750 

5.266 

4.750 

4.140 

3.502 

.0046 

12.556 

11.448 

10.708 

9.752 

7.84 

5.324 

4.802 

4.186 

3.54 

.0047 

12.70 

11.580 

10.830 

9.864 

7.928 

5.382 

4.854 

4.232 

3.578 

.0048 

12.842 

11.708 

10.952 

9.974 

8.018 

5.438 

4.904 

4.276 

3.616 

.0049 

12.982 

11.838 

11.072 

10.084 

8.106 

5.494 

4.956 

4.320 

3.654 

.005 

13.122 

11.964 

11.190 

10.192 

8.192 

5.550 

5.006 

4.364 

3.690 

.0051 

13.26 

12.090 

11.308 

10.300 

8.280 

5.604 

5.056 

4.408 

3.628 

.0052 

13.396 

12.216 

11.426 

10.406 

8.364 

5.660 

5.106 

4.452 

3.764 

.0053 

13.534 

12.34 

11.542 

10.512 

8.450 

5.714 

5.154 

4.494 

3.80 

.0054 

13.664 

12.46 

11.654 

10.614 

8.530 

5.768 

5.202 

4.536 

3.834 

.0055 

13.802 

12.584 

11.770 

10.720 

8.616 

5.82 

5.250 

4.578 

3.872 

.0056 

13.934 

12.704 

11.882 

10.824 

8.700 

5.874 

5.298 

4.620 

3.906 

.0057 

14.064 

12.824 

11.994 

10.924 

8.782 

5.924 

5.344 

4.660 

3.94 

.0058 

14.194 

12.942 

12.106 

11.026 

8.862 

5.978 

5.390 

4.70 

3.974 

.0059 

14.324 

13.060 

12.216 

11.126 

8.942 

6.028 

5.438 

4.74 

4.010 

.006 

14.452 

13.176 

12.324 

11.226 

9.022 

6.080 

5.484 

4.78 

4.044 

.0061 

14.578 

13.392 

12.434 

11.324 

9.102 

6.130 

5.530 

4.82 

4.076 

.0062 

14.704 

13.408 

12.54 

11.422 

9.180 

6.180 

5.574 

4.86 

4.110 

.0063 

14.830 

13.522 

12.648 

11.518 

9.26 

6.230 

5.620 

4.90 

4.144 

.0064 

14.954 

13.636 

12.754 

11.616 

9.336 

6.28 

5.664 

4.938 

4.176 

.0065 

15.078 

13.746 

12.858 

11.712 

9.414 

6.328 

5.708 

4.976 

4.208 

.0066 

15.2 

13.860 

12.962 

11.806 

9.490 

6.376 

5.752 

5.014 

4.240 

OPEN    CONDUITS 


165 


TABLE  I.  —  Continued. 


Sine  of 
the 
Slope 

a=FT8 

a=1.0 

m  = 
0.95 

m  = 
0.80 

m  = 
0.70 

w  = 
0.57 

772,= 

0.30 

m  = 
0.0 

K  = 

1.2 

K  = 
1.5 

K  = 
1.93 

.0067 

15.32 

13.970 

13.066 

11.902 

9.566 

6.424 

5.794 

5.052 

4.272 

.0068 

15.442 

14.080 

13.170 

11.994 

9.64 

6.472 

5.838 

5.090 

4.304 

.0069 

15.562 

14.190 

13.272 

12.088 

9.716 

6.520 

5.88 

5.128 

4.336 

.0070 

15.680 

14.298 

13.372 

12.18 

9.790 

6.566 

5.924 

5.164 

4.368 

.0071 

15.798 

14.404 

13.474 

12.272 

9.864 

6.614 

5.966 

5.202 

4.398 

.0072 

15.916 

14.512 

13.574 

12.362 

9.938 

6.660 

6.008 

5.238 

4.430 

.0073 

16.034 

14.618 

13.674 

12.454 

10.010 

6.706 

6.048 

5.274 

4.460 

.0074 

16.150 

14.724 

13.772 

12.542 

10.082 

6.752 

6.090 

5.310 

4.490 

.0075 

16.264 

14.828 

13.870 

12.654 

10.154 

6.798 

6.134 

5.346 

4.522 

.0076 

16.378 

14.934 

13.968 

12.722 

10.226 

6.842 

6.172 

5.380 

4.550 

.0077 

16.492 

15.038 

14.066 

12.810 

10.296 

6.888 

6.212 

5.416 

4.580 

.0078 

16.606 

15.142 

14.162 

12.898 

10.368 

6.932 

6.252 

5.452 

4.610 

.0079 

16.718 

15.244 

14.258 

12.986 

10.438 

6.976 

1  6.292 

5.486 

4.640 

.008 

16.830 

15.344 

14.352 

13.072 

10.508 

7.020 

6.332 

5.520 

4.668 

.00825 

17.106 

15.598 

14.588 

13.288 

10.680 

7.130 

6.430 

5.606 

4.740 

.0085 

17.378 

15.846 

14.820 

13.514 

10.850 

7.236 

6.528 

5.690 

.4.812 

.00875 

17.648 

16.090 

15.050 

13.706 

11.018 

7.342 

6.622 

5.774 

4.822 

.009 

17.914 

16.334 

15.278 

13.914 

11.184 

7.446 

6.716 

5.856 

4.952 

.00925 

18.174 

16.572 

15.498 

14.116 

11.346 

7.548 

6.810 

5.936 

5.020 

.0095 

18.432 

16.806 

15.720 

14.318 

11.508 

7.650 

6.900 

6.016 

5.088 

.00975 

18.688 

17.04 

15.938 

14.516 

11.668 

7.750 

6.990 

6.096 

5.154 

.01 

18.94 

17.270 

16.152 

14.712 

11.826 

7.848 

7.080 

6.172 

5.220 

.01025 

19.198 

17.496 

16.364 

14.906 

11.980 

7.940 

7.162 

6.244 

5.280 

.0105 

19.436 

17.72 

16.576 

15.096 

12.134 

8.042 

7.254 

6.326 

5.348 

.01075 

19.678 

17.942 

16.782 

15.286 

12.286 

8.138 

7.340 

6.400 

5.412 

.011 

19.920 

18.164 

16.988 

15.472 

12.436 

8.232 

7.426 

6.474 

5.474 

.01125 

20.158 

18.376 

17.192 

15.658 

12.586 

8.326 

7.510 

6.548 

5.536 

.0115 

20.394 

18.596 

17.392 

15.842 

12.734 

8.416 

7.592 

6.620 

5.598 

.01175 

20.628 

18.408 

17.592 

16.022 

12.878 

8.508 

7.674 

6.690 

5.658 

.012 

20.860 

19.020 

17.770 

16.202 

13.022 

8.578 

7.738 

6.762 

5.718 

.01225 

21.088 

19.228 

17.984 

16.380 

13.166 

8.686 

7.836 

6.830 

5.776 

.0125 

21.316 

19.434 

18.178 

16.556 

13.306 

8.776 

7.916 

6.900 

5.836 

.01275 

21.544 

19.644 

18.374 

16.734 

13.452 

8.862 

7.994 

6.970 

5.894 

.013 

21.762 

19.842 

18.560 

16.904 

13.586 

8.922 

8.048 

7.018 

5.934 

.01325 

21.982 

20.044 

18.748 

17.074 

13.724 

9.034 

8.148 

7.104 

6.008 

.0135 

22.20 

20.242 

18.934 

17.244 

13.862 

9.120 

8.226 

7.172 

6.064 

.01375 

22.418 

20.440 

19.118 

17.412 

13.996 

9.204 

8.302 

7.238 

6.120 

.014 

22.632 

20.636 

19.304 

17.580 

14.130 

9.286 

8.376 

7.304 

6.176 

.01425 

22.846 

20.830 

19.484 

17.746 

14.264 

9.370 

8.452 

7.368 

6.232 

.0145 

23.056 

21.022 

19.664 

17.908 

14.396 

9.452 

8.526 

7.432 

6.286 

.01475 

23.266 

21.214 

19.842 

18.072 

14.526 

9.532 

8.598 

7.496 

6.340 

.015 

23.474 

21.404 

20.02 

18.234 

14.656 

9.612 

8.672 

7.560 

6.392 

.01525 

23.680 

21.592 

20.196 

18.394 

14.784 

9.690 

8.740 

7.620 

6.444 

.0155 

23.886 

21.780 

20.370 

18.552 

14.912 

9.772 

8.814 

7.684 

6.498 

.01575 

24.050 

21.962 

20.544 

18.746 

15.040 

9.850 

8.884 

7.746 

6.550 

.016 

24.460 

22.146 

20.714 

18.866 

15.164 

9.930 

8.958 

7.810 

6.604 

.01625 

24.548 

22.330 

20.886 

19.024 

15.290 

10.006 

9.026 

7.868 

6.654 

.0165 

24.690 

22.512 

21.044 

19.178 

15.414 

10.082 

9.094 

7.928 

6.704 

.01675 

24.886 

22.692 

21.224 

19.330 

15.538 

10.160 

9.164 

7.990 

6.758 

166 


THE    FLOW  OF   WATER 
TABLE  I.  — Concluded. 


Sine  of 
the 
Slope 

a=  V™ 

a=1.0 

m  = 
0.95 

m  = 
0.80 

771  = 

0.70 

771  = 

0.57 

771  = 

0.30 

771  = 
0.0 

K  = 
1.2 

K  = 

1.5 

K  = 
1.93 

.017 

25.08 

22.870 

21.392 

19.484 

15.660 

10.236 

9.234 

8.050 

6.808 

.01725 

25.28 

23.048 

21.558 

19.634 

15.782 

10.308 

9.298 

8.106 

6.856 

.0175 

25.48 

23.224 

21.722 

19.784 

15.902 

10.382 

9.366 

8.166 

6.906 

.01775 

25.66 

23.400 

21.886 

19.934 

16.022 

10.456 

9.432 

8.224 

6.954 

.018 

25.84 

23.572 

22.048 

20.082 

16.142 

10.530 

9.498 

8.282 

7.002 

.01825 

26.04 

23.746 

22.210 

20.230 

16.260 

10.604 

9.564 

8.338 

7.052 

.0185 

26.22 

23.918 

22.370 

20.376 

16.378 

10.676 

9.630 

8.396 

7.098 

.01875 

26.42 

24.088 

22.430 

20.520 

16.494 

10.748 

9.694 

8.452 

7.148 

.019 

26.66 

24.314 

22.688 

20.664 

16.610 

10.818 

9.758 

8.508 

7.196 

.01925 

26.80 

24.426 

22.846 

20.808 

16.726 

10.890 

9.822 

8.564 

7.242 

.0195 

26.96 

24.594 

23.004 

20.950 

16.840 

10.960 

9.886 

8.620 

7.288 

.01975 

27.16 

24.760 

23.158 

21.092 

16.954 

11.030 

9.950 

8.674 

7.336 

.020 

27.34 

24.92 

23.314 

21.232 

17.068 

11.10 

10.012 

8.730 

7.382 

.0205 

27.70 

25.24 

23.618 

21.512 

17.390 

11.238 

10.136 

8.838 

7.474 

.021 

28.06 

25.58 

23.924 

21.790 

17.514 

11.374 

10.258 

8.944 

7.564 

.0215 

28.40 

25.  .90 

24.224 

22.062 

17.734 

11.508 

10.382 

9.050 

7.654 

.022 

28.76 

26.22 

24.526 

22.338 

17.954 

11.642 

10.502 

9.156 

7.742 

.0225 

29.12 

26.54 

24.830 

22.616 

18".  178 

11.774 

10.626 

9.260 

7.830 

.023 

29.44 

26.84 

25.10 

22.864 

18.378 

11.904 

10.738 

9.360 

7.916 

.0235 

29.78 

27.14 

25.40 

23.126 

18.588 

12.032 

10.854 

9.462 

8.002 

.024 

30.10 

27.44 

25.68 

23.386 

18.796 

12.160 

10.968 

9.562 

8.086 

.0245 

30.44 

27.76 

25.96 

23.642 

19-.  002 

12.246 

11.082 

9.662 

8.170 

.025 

30.76 

28.06 

26.24 

23.896 

19.208 

12.410 

11.194 

9.760 

8.254 

.030 

33.88 

30.90 

28.90 

26.32 

21.154 

13.594 

12.262 

10.690 

8.970 

.040 

39.46 

35.98 

33.66 

30.64 

24.634 

15.702 

14.164 

12.348 

10.442 

.050 

44.40 

40.48 

37.96 

34.50 

27.62 

17.550 

15.830 

13.802 

11.672 

.060 

48.90 

44.60 

41.70 

37.98 

30.54 

19.226 

17.342 

15.118 

12.786 

.070 

53.06 

48.38 

45.26 

41.22 

33.14 

20.766 

18.732 

16.330 

13.808 

.080 

56.96 

51.92 

48.56 

44.24 

35.56 

22.20 

20.024 

17.458 

14.764 

.090 

60.62 

55.26 

52.90 

44.96 

37.84 

23.524 

21.242 

18.518 

15.662 

0.100 

64.08 

58.44 

54.66 

49.78 

40.0 

24.82 

22.388 

19.526 

16.706 

WEIR  DISCHARGES  167 

Weir  Discharges. 

FRANCIS'  FORMULA. 

The  discharge  of  a  sharp-edged  measuring  weir  is  usually 
computed  from  Francis'  formula,  which  reads: 

Q  =3.33  (b-nQ.lH)  (H  +  h)*  -  M, 

in  which  Q  =  discharge  in  cubic  feet  per  second; 
b  =  breadth  of  weir  in  feet; 
n  =  number  of  end  contractions; 
H  =  the  vertical  distance  between  the  crest  of  the 
weir  and  the  surface  of  the  still  water  in  the 
reservoir  or  the  channel; 

h  =  the  head  due  to  the  velocity  of  approach. 

The  head  due  to  the  velocity  of  approach  is  found  from  the 
equation 

- 


in  which  Q  =  discharge  found  from  the  formula  given  above, 
neglecting  the  velocity  of  approach; 

A  =  cross-section  of  the  channel  or  reservoir  parallel 
to  the  weir,  at  the  point  where  the  surface 
of  the  water  begins  to  slope  towards  the  weir. 

If  the  discharge  of  the  weir  is  small  in  comparison  with  the 
width  and  depth  of  the  channel  or  the  contents  of  the  reser- 
voir the  velocity  of  approach  and  the  head  due  to  it  may  be 
neglected. 

Table  K  contains  values  of  3.33  #1 
The  table  is  used  as  follows: 

Let  the  depth  of  the  water  from  the  crest  of  the  weir  to  the 
still  surface  be  3  feet. 


168 


THE   FLOW   OF  WATER 


Let  the  head  due  to  the  velocity  of  approach  be  0.1  foot. 
Then  3.33  (H  +  h)$  -  3.33  ft*  =  3.33  (3.1)*  -  3.3  (0.1)* 
=  18.176  -  0.1053  =  18.0707. 

Let  the  breadth  of  the  weir  be  10  feet  and  we  have: 

Q  =  (io  -  2  X  0.3)  X  18.0707  =  169.86458    cubic    feet    per 
second. 

TABLE  K. 


H 

3.33  H% 

H 

3.33  H* 

H 

3.33  H* 

H 

3.33  #2 

H 

3.33  H^ 

0.01 

0.00333 

0.30 

0.5472 

0.78 

2.294 

1.65 

7.025 

2.85 

16.022 

0.02 

0.009406 

0.32 

0.6028 

0.80 

2.383 

1.70 

7.381 

2.90 

16.445 

0.03 

0.01722 

0.34 

0.6602 

0.82 

2.478 

1.75 

7.709 

2.95 

16.872 

0.04 

0.02664 

0.36 

0.7193 

0.84 

2.564 

1.80 

8.042 

3.0 

17.307 

0.05 

0.03898 

0.38 

0.7792 

0.86 

2.656 

1.85 

8.379 

3.05 

17.736 

0.06 

0.05125 

0.40 

0.8425 

0.88 

2.749 

1.90 

8.721 

3.10 

18.177 

0.07 

0.06167 

0.42 

0.9064 

0.90 

2.843 

1.95 

9.068 

3.15 

18.61 

0.08 

0.07535 

0.44 

0.9719 

0.92 

2.938 

2.0 

9.418 

3.20 

19.06 

0.09 

0.08991- 

0.46 

1.0389 

0.94 

3.035 

2.05 

9.774 

3.25 

19.50 

0.10 

0.1053 

0.48 

1.1074 

0.96 

3.132 

2.10 

10.144 

3.30 

19.96 

0.11 

0.1215 

0.50 

1.1773 

0.98 

3.231 

2.15 

10.498 

3.35 

20.42 

0.12 

0.1383 

0.52 

1.2486 

1.0 

3.330 

2.20 

10.866 

3.40 

20.88 

0.13 

0.1561 

0.54 

1.3215 

.05 

3.583 

2.25 

11.239 

3.45 

21.34 

0.14 

0.1744 

0.56 

1.3958 

.10 

3.842 

2.30 

11.616 

3.50 

21.81 

0.15 

0.1934 

0.58 

1  .  4708 

.15 

4.107 

2.35 

11.996 

3.55 

22.27 

0.16 

0.2133 

0.60 

1.5476 

.20 

4.377 

2.40 

12.381 

3.60 

22.75 

0.17 

0.2333 

0.62 

1.6260 

.25 

4.654 

2.45 

12.770 

3.65 

23.22 

0.18 

0.2543 

0.64 

1  .  7050 

.30 

4.936 

2.50 

13.163 

3.70 

23.70 

0.19 

0.2758 

0.66 

1.7855 

.35 

5.223 

2.55 

13.560 

3.75 

24.18 

0.20 

0.2978 

0.68 

1.8672 

.40 

5.516 

2.60 

13.960 

3.80 

24.67 

0.22 

0.3436 

0.70 

1.7502 

.45 

5.815 

2.65 

14.365 

3.85 

25.15 

0.24 

0.3915 

0.72 

2.0344 

.50 

6.117 

2.70 

14.773 

3.90 

25.65 

0.26 

0.4413 

0.74 

2.1197 

.55 

6.426 

2.75 

15.186 

3.95 

26.14 

0.28 

0.4938 

0.76 

2.206 

.60 

6.739 

2.80 

15.602 

4.0 

26.64 

THE   FORMULA  OF  BAZIN. 

The  weir  formula  of  Francis  is  based  on  experiments  made 
with  heads  ranging  between  5  and  19  inches  and  with  weir  crests 
up  to  10  feet  in  length. 

The  accuracy  of  the  formula  when  applied  to  flow  over  weirs 
having  end  contractions  has  been  demonstrated;  it  also  gives 
fairly  good  results  when  applied  to  flow  over  weirs  whose  sides 
are  flush  with  the  walls  of  the  channel  of  approach.  In  that 
case  n  =  0.  The  difficulty  in  the  application  of  the  formula  of 


WEIR   DISCHARGES  169 

Francis  consists  in  the  fact,  that  it  is  frequently  impossible  to 
evaluate  properly  the  head  due  to  the  velocity  of  approach. 

If  the  formula  of  Bazin  is  used  the  head  due  to  the  velocity  of 
approach  does  not  enter  directly  into  calculations;  it  is  replaced 
by  a  coefficient  which  depends  for  its  value  on  the  relation 
between  the  head  and  the  vertical  distance  between  the  crest  and 
the  floor  of  the  channel  of  approach. 

Bazin  conducted  his  experiments  with  weirs  0.5,  1.0  and  2.0 
meters  wide,  the  heads  ranging  between  0.05  meters  (2  inches) 
and  0.6  meters  (24  inches).  The  crests  of  the  weirs  were  raised 
to  various  heights  above  the  floors  of  the  channels  of  approach 
and  the  sides  were  flush  with  the  walls  of  the  channels.  The 
formula  of  Bazin  reads  : 


Q 


=  f  mil  +  0.55  (—  ^-Jl  Lh 


in  which  m  =  0.6075  + 


h  =  the  head  above  the  crest  to  the  surface  of  tks  still 

water; 
p  =  the  depth  of  the  water  below  the  crest  to  the  floor 

of  the  channel  of  approach. 

The  formula  as  given  holds  good  for  any  system  of  measure. 
For  English  measure  it  may  be  written  : 

Q  =  (3.2485  h*  +  0.07914  Vh)  L  j"l  +  0.55       h*      1  . 
Values  of   ~1  +    °'55   l    are  found   in  Table  L.a.     It  will 


, 

(p  +  /&) 

be  observed,  that  the  value  of  this  factor  diminishes  rapidly  as 

the  relation  -  diminishes  in  value. 
P 

It  is  equal  to  1.2444  for  -  =  f  .  1.0220  for  -  =  \  . 

pi  p      4 

1.1375  for  -  =  i  .  1.0068  for  -  =  |  . 

pi  p      8 

1.0611  for  -  =  i  . 

p      2 


170 


THE  FLOW  OF  WATER 


TABLE  L.a. 


; 

i  •  v 

1 

0.55  A2 

h  : 

0.55^ 

*  •  ft 

X 

(p+h)2 

(p+h)2 

25 

] 

.0008 

3 

1.0344 

20 

] 

[.0012 

2.75 

.0391 

15 

3 

L.0021 

2.5 

.0449 

10 

] 

L.0045 

2.25 

.0529 

9.5 

3 

L.0050 

2 

.0611 

9 

] 

L.0055 

1.75 

.0727 

8.5 

1 

L.0061 

1.5 

.0880 

8 

: 

L.0068 

1 

1.25 

.1086 

7.5 

.0076 

1 

1 

.1375 

7 

.0086 

1.25 

1 

.1698 

6.5 

.0097 

1.5 

1 

1.1979 

i 

6 

.0112 

1.25 

1 

1.2228 

i 

5.5 

.0130 

2 

1.2444 

i 

5 

.0153 

2.25 

1.2638 

i 

4.75 

.0166 

2.5 

1.2806 

. 

4.5 

.0181 

2.75 

1.2960 

4.25 

.0199 

3 

1.3095 

• 

; 

4 

.0220 

3.25 

1.3220 

i 

375 

.0244 

3.5 

1.3331 

i. 

3.5 

.0276 

3.75 

1.3431 

i 

3.25 

.0304 

4 

1  .  3520 

WEIR  DISCHARGES 


171 


TABLE  L.b. 
VALUES  OF  Q  =  3.2485  A!  +  0.07914  Vh. 


h 

Q 

In  Cu.  Ft. 
per  Sec. 

h 

Q 
In  Cu.  Ft. 
per  Sec. 

h 

Q 

In  Cu.  Ft. 
per  Sec. 

h 

Q 

In  Cu.  Ft. 
per  Sec. 

0.01 

0.0111 

0.82 

2.4900 

2.60 

13.7481 

4.65 

32.7513 

0.02 

0.0204 

0.84 

2.5738 

2.65 

14.1393 

4.70 

33.2832 

0.04 

0.0418 

0.86 

2.6646 

2.70 

14.5405 

4.75 

33.8031 

0.06 

0.0695 

0.88 

2.7565 

2.75 

14.9417 

4.80 

34.3340 

0.08 

0.0960 

0.90 

2.8493 

2.80 

15.3529 

4.85 

34.8749 

0.10 

0.1279 

0.92 

2.9432 

2.85 

15.7641 

4.90 

35.4158 

0.12 

0.1625 

0.94 

3.0380 

2.90 

16.1753 

4.95 

35.9561 

0.14 

0.1998 

0.96 

3.1338 

2.95 

16.5864 

5.0 

36.4976 

0.16 

0.2304 

0.98 

3.2308 

3.00 

17.0176 

5.10 

37.5894 

0.18 

0.2819 

1.0 

3.3276 

3.05 

17.4387 

5.2 

38.7011 

0.20 

0.3261 

1.05 

3.5763 

3.10 

17.8698 

5.3 

39.8229 

0.22 

0.3724 

1.10 

3.8313 

3.15 

18.3010 

5.4 

40.9446 

0.24 

0.4208 

1.15 

4.0911 

3.20 

18.7421 

5.5 

42.0863 

0.26 

0.4712 

1.20 

4.3570 

3.25 

19.1732 

5.6 

43.2380 

0.28 

0.5238 

1.25 

4.6288 

3.30 

19.6143 

5.7 

44.3996 

0.30 

0.5773 

1.30 

4.9065 

3.35 

20.0654 

5.8 

45.5713 

0.32 

0.6330 

1.35 

5.1873 

3.40 

20.5165 

5.9 

46.7530 

0.34 

0.6904 

1.40 

5.4750 

3.45 

20.9675 

6.0 

47.9440 

0.36 

0.7493 

.45 

5.7676 

3.50 

21.4186 

6.2 

50.3478 

0.38 

0.8090 

.50 

6.0653 

3.55 

21.8797 

6.4 

52.8010 

0.40 

0.8720 

.55 

6.3682 

3.60 

22.3407 

6.6 

55.2940 

0.42 

0.9356 

.60 

6.6755 

3.65 

22.8018 

6.8 

57.8071 

0.44 

.0080 

.65 

6.9560 

3.70 

23.2728 

7.0 

60.3701 

0.46 

.0673 

.70 

7.3046 

3.75 

23.7438 

7.2 

62.9731 

0.48 

.1353 

1.75 

7.6261 

3.80 

24.2248 

7.4 

65.6061 

0.50 

.2047 

1.80 

7.9515 

3.85 

24.6959 

7.6 

68.2791 

0.52 

.2753 

1.85 

8.2820 

3.90 

25.1769 

7.8 

70.9918 

0.54 

.3473 

1.90 

8.6175 

3.95 

25.6598 

8.0 

73.7341 

0.56 

.4204 

1.95 

8.9569 

4.00 

26.1463 

8.2 

.76.5075 

0.58 

.4955 

2.0 

9.3023 

4.05 

26.6398 

8.4 

79.3202 

0.60 

.5715 

2.05 

9.6483 

4.10 

27.1308 

8.6 

82.1629 

0.62 

.6485 

2.10 

10.0011 

4.15 

27.6218 

8.8 

85.0157 

0.64 

.7265 

2.15 

10.3575 

4.20 

28.1228 

9.0 

87.9483 

0.66 

.8065 

2.20 

10.7188 

4.25 

28.6237 

9.2 

90.8909 

0.68 

.8865 

2.25 

11.0830 

4.30 

29.1347 

9.4 

94.0335 

0.70 

.9694 

2.30 

11.4524 

4.35 

29.6357 

9.6 

96.8661 

0.72 

2.0524 

2.35 

11.8247 

4.40 

30.1466 

9.8 

99.9187 

0.74 

2.1363 

2.40 

12.2010 

4.45 

30.6675 

10.0 

102.9812 

0.76 

2.2212 

2.45 

12.5823 

4.50 

31.1785 

0.78 

2.3081 

2.50 

12.9656 

4.55 

31.6994 

0.80 

2.3950 

2.55 

13.3569 

4.60 

32.2204 

172 


THE   FLOW   OF   WATER 


If  accurate  results  are  desired  from  the  application  of  this 
formula  the  depth  p  as  well  as  the  length  L  should  never  be  less 
than  2  h,  and  the  width  of  the  channel  of  approach  should 
increase  up  stream  from  the  crest. 

Heads  are  most  conveniently  and  accurately  ascertained  by 
means  of  a  plumb-bob,  the  string  of  which  is  hung  over  a  nail 
driven  horizontally  and  pulled  horizontally  along  a  board  to 
which  a  graduated  scale  is  attached.  A  datum  reading  is  taken 
and  laid  off  on  the  scale  when  the  surface  of  the  water  is  just 
flush  with  the  crest  and  the  point  of  the  plumb-bob  grazes  the 
surface  when  it  is  made  to  swing  to  and  fro. 

Of  weirs  not  originally  constructed  to  be  measuring  devices 
those  most  frequently  found  are  the  sharp-crested  triangular 
weir,  the  triangular  weir  with  a  quarter  round  crest  and  the 
rectangular,  broad-crested  weir.  The  factors  of  proportional 
discharge  for  these  shapes,  for  which  we  are  indebted  to  Bazin, 
the  Cornell  Engineers,  G.  W.  Rafter  and  others,  are  as  follows, 
the  down  stream  face  being  in  all  cases  vertical,  air  admitted 
under  the  descending  sheet  of  water  and  the  relation  between  h 
and  p  being  the  same  as  for  the  sharp  edged  measuring  weir : 


Head  on  Sill  in  Feet. 

Description  of  Shape  of  Cross-Section 

0.5 

1.0 

1.5 

2.0 

3.0 

4.0 

Crest  triangular,  up  stream  face 

inclined  1:1. 

1.06 

1.079 

1.092 

1.094 

1.082 

1.072 

Crest  quarter  round  of  circle,  di- 

ameter 1  meter,  up  stream  face 

inclined  1:1. 

0.971 

0.983 

1.012 

1.040 

1.072 

1.097 

Crest  rectangular,  both  faces  ver- 

tical.    Thickness  of  wall: 

0  5  feet 

0  902 

0  972 

1  0 

1  0 

1  0 

1.0 

1.0  feet          .    .   . 

0  830 

0  904 

0  957 

0  989 

1  0 

1.0 

1.5  feet  

0  819 

0  879 

0  910 

0  925 

0  928 

0.947 

3.0  feet  

0  797 

0  812 

0  821 

0  821 

0  813 

0.808 

6.0  feet  

0  785 

0  800 

0  807 

0  805 

0  796 

0.790 

9  0  feet      .       ... 

0  783 

0  798 

0  803 

0  800 

0  797 

0  783 

16.0  feet  

0.783 

0.792 

0.797 

0.797 

0.784 

0.777 

WEIR  FORMULAE  173 

Weir  Formulae. 

Weirs  are  constructed  for  the  following  purposes: 

(1)  To  measure  the  discharge  of  .a  conduit. 

(2)  To  regulate  the  discharge  of  a  conduit. 

(3)  To  serve  as  impounding  and  regulating  dams  for  the 
storage  of  water. 

(4)  To  raise  the  surface  of  the  water  at  a  certain  point  to  a 
certain  level. 

According  to  the  manner  of  outflow  weirs  are  classified  as 
follows : 

(1)  Complete  overflow  weirs,  when  the  crest  of  the  weir  is 
above  the  surface  of  the  run-off  water. 

(2)  Incomplete  weirs,  when  the  crest  of  the  weir  is  below  the 
surface  of  the  runoff  water. 

(3)  Discontinuous    weirs    (wing   dams,    bridge    piers,    etc.) 
when  the  weir  does  not  extend  the  whole  width  of  the  channel. 

(4)  Sluice    weirs    (water-gate,    head-gate,    regulating    weir, 
needle  weir,  etc.),  when  the  water  flows  out  through  an  orifice. 

Theoretically   the   discharge   through   a   rectangular   sharp- 
edged  orifice  is  found  as  follows: 

Let  b   be  the  breadth  of  a  rectangular  jet, 
h^  the  depth  of  its  upper, 

h2  the  depth  of  its  lower  surface  below  the  surface  of  the 
still  water  (Fig.  8). 

An  infinitesimal  thin  layer  of  the  jet  between  its  surface  and 
an  infinitesimal  depth  h  has  a  section  equal  to  bdh. 
The  velocity  of  flow  in  this  infinitesimal  layer  bdh  is  equal  to 

v  =  V2gh. 
The  discharge  will  consequently  be 


r 

Jh! 


bV2ghdk, 

which  integrated,  gives  for  the  discharge  of  the  whole  jet 


174 


THE    FLOW   OF   WATER 


Let  B  be  the  breadth  of  the  orifice, 

Hl  the  depth  of  its  upper, 

H2  the  depth  of  its  lower  edge  below  the  free  surface, 
we  then  have  for  the  coefficient  of  discharge 


B  (H*  -  ff,*) 

and  for  the  discharge  in  terms  of  the  orifice 


(1) 


FIG. 


The  value  of  C,  the  coefficient  of  discharge,  differs  with  the 
nature  of  the  orifice,  and  must  be  found  by  experiment.  (For 
sharp-edged  orifices  and  weirs  C  =  0.622,  for  broad-crested 
weirs  C  =  0.577). 

For  the  discharge  through  a  rectangular  sharp-edged  notch  we 
have,  since  there  is  in  this  case  no  head  Hl 

Q  =  $CBV?rg  H,t.  (2) 

The  discharge  through  rectangular  notches  and  over  sharp- 
crested  weirs,  has  been  minutely  investigated  by  Bazin,  Francis, 
and  others.  Bazin  found  for  the  discharge  the  expression, 


(3) 


in  which        c  =  0 . 6075  + 


0.0148 
h 


p  =  height  of  crest  above  bottom  of  channel. 


WEIR   FORMULAE  175 

Francis  found  that  the  loss  of  discharge  due  to  end  contraction 
is  equal  to  •£$  the  height  of  the  submerged  opening  for  each 
contraction.  The  discharge  is  consequently 

Q  =  f  c  (b  -  0.1  mH)  V2g  H?, 

in  which  m  is  the  number  of  end  contractions. 

If  Francis'  formula  is  used  and  the  discharge  is  relatively 
large  compared  with  the  dimensions  of  the  conduit,  the  head 
due  to  the  velocity  of  approach  must  also  be  considered.  This 
head  is  equal  to 


the  velocity  of  approach  is  equal  to 


b0  and  h0  being  the  breadth  and  depth  of  the  channel  at  the 
point  where  the  surface  of  the  water  begins  to  drop  towards  the 
crest  of  the  opening.  Making  this  correction  for  the  head  due 
to  the  velocity  of  approach,  Francis'  formula  becomes 

Q  =  |C(6  -0.1H)V2g[(H  +  a)*  -a*], 
in  which  c  =  0.622  for  sharp-edged  orifices. 
Putting  f  0.622  V2g  =  3.33,  Francis'  formula  reads 

Q  =  3.33  (6  -  raO.l  H)  [(H  +  a)1  -  a*].  (4) 

Bazin's  and  Francis'  formulae  give  equally  good  results;  the 
latter  is  the  one  most  frequently  used  in  this  country. 

To  measure  the  discharge  of  a  small  stream  (pipe  line,  flume, 
etc.),  a  temporary  weir  of  planks  is  usually  constructed.  In 
order  to  arrive  at  accurate  results  care  must  be  taken  that  the 
sill  or  the  crest  of  the  weir  is  perfectly  level,  that  it  is  at  right 
angles  to  the  line  of  flow,  and  that  it  is  above  the  surface  of  the 
run-off  water.  The  head  on  the  sill  should  not  be  less  than  one- 
half,  nor  more  than  2  feet,  and  the  depth  of  water  in  the  channel 
should  be  at  least  three  times  the  head  on  the  crest.  In  order 
to  measure  the  head  on  the  sill  a  stake  is  driven  in  the  bed  of 
the  stream  a  short  distance  above  the  weir.  The  top  of  the 


176 


THE   FLOW  OF  WATER 


stake  must  be  on  a  perfect  level  with  the  crest  of  the  weir.  A 
thin-edged  graduated  scale  fastened  vertically  to  the  top  of  the 
stake  is  very  convenient.  On  this  gauge  the  height  H,  the 
head  on  the  sill  is  read  off  to  the  surface  of  the  still  water.  , 


FIG.  9.     Measuring  Weir. 

Weirs  intended  to  regulate  the  discharge  of  a  conduit,  or  to 
raise  the  surface  of  the  water  at  a  certain  point  to  a  certain 
level  are  constructed  of  various  materials  and  in  various  forms. 
A  complete  overflow  weir,  when  constructed  of  masonry  in  the 
bed  of  a  stream,  is  usually  of  the  form  shown  in  Fig.  10. 

M 


FIG.  10.    Complete  Overflow  Weir. 

The  height  of  the  weir  necessary  to  raise  the  surface  of  the 
water  to  a  given  height  h,  is  found  as  follows : 


WEIR   FORMULAE 


177 


Let  H  be  the  head  on  the  sill,  measured  to  still  water, 
6  the  breadth  of  the  channel, 
a  the  height  due  to  the  velocity  of  approach, 
h  the  difference  of  level  between  the  surface  of  the  water 
down  stream  and  up  stream,  or  the  swell  (MO,  Fig.  10) 

and  the  discharge  is 

Q  =  %cbV2^[(H  +  d)*  -a*], 
from  which  we  find  for.  the  head  on  the  sill 

Q 


Denoting  the  height  of  the  weir  above  the  bottom  of  the 
channel  by  x  and  the  depth  of  the  run-off  water  down  stream 

by  /,  we  have 

x  +  H  =  h  +  f, 
hence  x  =  (/  +  h)  —  H, 


or 


when  the  velocity  of  approach  is  small. 


FIG.  11.     Incomplete  Weir. 

For  an  incomplete  weir  the  discharge  and  the  height  of  crest 

necessary  to  raise  the  water  to  a  given  level  are  found  as  follows : 

The  head  on  the  sill  (MN,  Fig.  11)  is  greater  than  the  swell- 


178 


THE  FLOW  OF  WATEK 


head  (MO),  therefore  only  the  water  above  0  flows  off  freely, 
while  the  water  below  0  flows  off  under  the  head  (MO)  =  h. 
The  discharge  through   (MO)  is 


and  that  through 


(ON)  =  H  -  his 


-h)V2g(h 


hence  the  whole  discharge: 

Q  =  cb  V2g%  [(h  +  a)*  -  a*]  +  H-hV(h  +  a). 
From  the  discharge  Q  and  the  height  h  (MO)  to  which  the 
water  is  raised,  we  find  for  the  height  of  water  above  the  crest, 
or  the  head  on  the  sill, 


H  = 


Q 


(h 


-*. 


hence  we  have  for  the  necessary  height  of  the  crest  of  the  weir 
above  the  bottom  of  the  channel  (NP), 

(NP)  =  x=(f  +  h)-H  =  [(OP)  +  (MO)]  -  (MN). 

Neglecting  the  velocity  of  approach  we  have  for  the  height  of 
the  weir  the  simple  expression 


(NP)  = 


Q 


cb  V2  gh 


FIG.  12.    Discontinuous  Weir. 


WEIR  FORMULAE 


179 


Wing  dams  are  built  whenever  an  obstruction  extending  the 
whole  width  of  the  stream  is  either  on  account  of  navigation 
not  permissible,  or  on  account  of  the  form  of  cross-section  of 
the  channel  not  feasible  or  necessary. 


FIG.  13.     Wing  Dam. 

While  for  overflow  weirs  the  usual  problem  consists  in  finding 
the  height  of  sill  or  crest  necessary  to  raise  the  water  to  a  given 
level,  the  problem  for  wing  dams  consists  in  finding  the  breadth 
of  channel  required  to  be  closed  in  order  to  raise  the  surface  of 
the  water  to  the  given  level. 

Let        QR  =  b  =  breadth  of  efflux  (Fig.  13), 
M  N  =  h  =  the  height  of  swell, 
NO  =  }  =  the  undercurrent, 

and  we  have  for  the  quantity  of  water  flowing  off  freely  above  / 
the  undercurrent 


and   for  the  undercurrent   / 

q2= 
the  whole  discharge  is  consequently 


from  which  We  find,  for  the  breadth  of  efflux, 

Q 


b  = 


c  (f  ft  +  /)  V2gh 


180 


THE  FLOW    OF   WATER 


If  the  velocity  of  flow  in  the  stream  is  great,  or  the  swell  h 
comparatively  small,  it  will  be  necessary  to  consider  the  velocity 
of  approach.  Denoting  as  before  by  a  the  head  due  to  the 
velocity  of  approach,  we  have  for  the  water  flowing  off  freely 


and  for  the  undercurrent 


q2  =  cbf  V2g  (h  +  a), 
for  the  whole  discharge 

Q  = 

and  finally  for  the  breadth  of  efflux 

Q 


b  = 


c  V20  (f  [(h  +  a)*  -  at]  +  /  \//i  4-  a 

This  formula  may  be  applied  to  discontinuous  weirs  of  any 
description,  such  as  bridge  piers,  etc.,  etc.     Denoting  by  b  the 


FIG.  14.    Sluice  Weir. 


sum  of  the  openings  between  bridge  piers  the  swell  may  for 
instance  be  found  by  putting 


swell  =  h 


\cb 


Wi 


The  coefficient  of  efflux  for  discontinuous  weirs  is  very  high, 
usually  only  the  end  contraction  needs  to  be  considered. 

For  wing  dams  c  =  0.98  will  give  good  results.  For  well- 
rounded  bridge  piers  c  may  be  taken  equal  to  0.90,  for  those 


WEIR   FORMULAE 


181 


forming  acute  angles  c  =  0.95,  and  for  those  of  elliptical  cross- 
section  c  =  0.97. 

Sluice  weirs  are  constructed  to  regulate  the  discharge  of  a 
conduit  or  reservoir  as  well  as  to  raise  the  surface  of  the  water 
to  a  given  level. 

In  computations  of  the  discharge  of  sluice  weirs,  the  head  H 
is  measured  from  the  free  surface  to  the  center  of  the  opening. 

If  the  water  flows  off  freely  we  have  for  the  discharge 

Q  =  cfb  V2~ 
and 
in  which 


and 


H  ~2~gW 

f  is  the  height  of  the  opening, 
b  =  the  breadth 
c  =  0.60. 


FIG.  15. 

For  a  given  discharge  and  a  given  head  H  the  height  of  the 
opening  is  given  by 

£    _ 


Q 


cbV2gH 

In  case  the  surface  of  the  run-off  water  down  stream  rises 
above  the  sluice  opening,  the  effective  head  reduces  to  the  dis- 
tance MN  (Fig.  15)  and  we  have  for  the  height  of  opening 

Q 


/  = 


cb  V2g  (MN) 


182 


THE   FLOW   OF    WATEK 


If  as  in  Fig.  16  the  surface  of  the  run-off  water  downstream 
lies  somewhere  within  the  opening,  a  part  of  the  water  runs  off 
under  water,  while  the  rest  flows  off  freely. 

Let  MO  -  H 

NO -ft 

OP  =  /„ 
and  the  discharge  through  NO  is 


3l  -  cfj>  V2gH  - 
and  the  discharge  through  OP 


q2  =  cf2b  V 
therefore  the  whole  discharges  through   NP 

Q  =  cb  V2g  (f1  VH  -0.5/!  +  /2  Vh). 


m 


FIG.  16. 

For  a  given  discharge  Q,  a  given  effective  head  H(MO)  and 
a  given  height  /2  of  the  sill  below  the  surface  of  the  run-off  water, 
the  height  fiy  or  the  distance  of  the  lower  edge  of  the  sluice  board 
above  the  surface  of  the  runoff  water  may  be  found  by  putting 

Q 


LOSS   OF   HEAD  183 

METHODS    OF    MEASUREMENT. 

Loss  of  Head. 

A. 

When  a  conduit  discharges  into  an  open  tank  or  reservoir, 
or  into  the  open  air,  the  loss  of  head  is  ascertained  by  levelling 
between  the  surface  of  the  source  of  supply,  and  the  surface  of 
the  discharge  tank,  reservoir  or  outflowing  stream.  When 
this  is  not  the  case,  or  when  the  loss  in  part  of  the  conduit  only 
is  to  be  .found,  other  methods  must  be  employed.  Where  the 
pressures  are  not  great,  open  stand  pipes  or  piezometers  are  most 
convenient,  otherwise  the  pressures  are  measured  by  means  of 
manometers.  A  mercury  manometer  of  the  form  generally 
used  has  the  following  essentials:  A  cast-iron  mercury  reservoir 
into  one  side  of  which  a  glass  plate  is  fitted  through  which  the 
height  of  the  mercury  within  may  be  observed.  A  metal  tube 
with  a  gate  valve  connects  the  top  of  the  reservoir  with  the  main 
pipe  at  the  point  at  which  the  pressure  is  to  be  measured.  At 
its  highest  point,  this  tube  has  an  air  valve.  Into  the  mercury 
reservoir,  which  is  about  half  filled  with  mercury,  a  vertical  tube 
is  placed,  nearly  reaching  to  the  bottom.  This  tube,  usually 
one  quarter  of  an  inch  in  diameter,  is  of  brass  or  wrought  iron  in 
its  lower  part  and  of  glass  in  its  upper  part.  To  the  glass  tube 
a  graduated  scale  is  attached.  As  mercury  is  very  sensitive  to 
changes  of  temperature  the  tube  is  surrounded  by  a  water- 
jacket,  in  its  upper  parts  also  of  glass.  When  the  gauge  is  to 
be  used  the  air  valve  in  the  connecting  tube  is  opened  and  also, 
by  degrees,  the  gate  valve.  When  the  air  is  wholly  removed 
the  air  valve  is  closed  and  the  gate  valve  fully  opened.  The 
pressure  of  the  water  in  the  reservoir  depresses  the  surface  of 
the  mercury  and  causes  it  to  rise  in  the  tube.  The  height  of  the 
mercury  column  above  the  surface  of  the  mercury  in  the  reservoir 
is  read  on  the  graduated  scale,  both  at  times  of  discharge  and 
times  of  no  discharge. 

If  a  is  the  difference  of  the  heights  of  the  mercury  columns  at 
two  sections  at  times  of  no  discharge,  and  A  the  difference  at 


184  THE   FLOW  OF  WATER 

times  of  discharge,  the  loss  of  head  between  the  two  sections 
whose  pressures  are  measured  is  equal  to 

H  =  13.6   A  -  a, 

13.6  being  the  specific  gravity  of  mercury. 

When  the  conduit  is  of  great  length  and  the  difference  between 
the  pressures  at  two  sections  considerable,  a  form  of  the  manom- 
eter known  as  the  Bourdon  gauge,  is  used  with  good  results.  The 
essential  parts  of  this  instrument,  universally  used  as  a  steam- 
gauge,  consist  of  a  hollow  curved  metal  spring,  one  end  of  which 
is  free  to  curve,  while  the  other  is  fastened  to  the  case  of  the 
instrument.  A  pipe  connects  the  interior  of  the  tube,  which 
is  oval  in  cross-section,  with  the  main  pipe  at  the  point  where 
the  pressure  is  to  be  measured.  The  pressure  of  the  liquid 
expands  the  spring,  the  free  end  moves  and  by  a  lever  the  move- 
ment is  transmitted  to  a  toothed  bar  lever,  which  again  transmits 
the  motion  to  a  toothed  wheel.  The  movement  of  the  spring, 
thus  converted  into  rotary  motion,  is,  by  a  pointer,  indicated 
upon  a  graduated  circular  scale.  The  pressure  is  indicated  in 
pounds  per  square  inch.  This  is  converted  into  feet  of  pressure 
by  dividing  it  by  0.434. 

If  A  is  the  difference  between  the  indicated  pressures  at  two 
sections  at  times  of  discharge  and  a  the  difference  at  times  of 
no  discharge,  the  loss  of  head  between  the  two  sections  is  equal 

to 

A  —  a 


H  = 


0.434 


Discharge  of  Conduits  under  Pressure. 
B. 

Discharges  are  measured  by  means  of  vessels,  tanks,  by  the 
rise  in  the  surface  of  a  reservoir,  or  the  overflowing  stream  is 
measured  by  a  weir,  an  orifice  or  the  current  meter.  When 
these  methods  are  not  feasible,  some  form  of  water  meter  is 
used.  The  best  known  devise  of  this  kind  is  the  Venturi  meter, 
invented  by  Herschel  and  named  for  a  celebrated  Italian 
hydraulician. 


DISCHARGE    OF   CONDUITS  UNDER   PRESSURE 


185 


The  theory  of  the  Venturi  meter  is  based  on  the  principles 
enunciated  by  Bernouilli: 

"The  fall  of  the  free  surface  level  between  two  sections  of  a 
conduit  is  equal  to  the  difference  of  the  heights  due  to  the 
velocities  at  the  sections." 

If  p±  is  the  pressure  at  one  section  of  a  conduit  and  vl  the 
velocity  and  p2  and  v2  the  pressure  and  velocity  at  another  sec- 
tion and  y  and  y:  elevations  above  datum,  then 


GG+y~ 


In  Fig.  17  the  line  pv  p2,  p3,  shows  the  theoretical  variation 
of  the  free  surface  level  due  to  the  contraction  and  subsequent 
enlargement  of  a  conduit.  The  line  pv  pv  p5,  shows  the  actual 
variation,  the  difference  being  due  to  the  pressure  expended 
in  overcoming  the  frictional  resistance  of  the  walls  of  the  con- 
duit. It  will  be  observed  that  this  difference  increases  with 
the  distance. 


FIG.  17. 


Differences  of  pressure  in  sections  of  conduits  not  far  apart 
are  most  conveniently  measured  by  mercury  difference  gauges. 
In  Fig.  18  is  shown  a  gauge  of  this  kind  connected  to  sections, 
the  "pressures  at  which  are  to  be  compared. 


186 


THE    FLOW   OF   WATER 


The  bottom  of  the  gauge  is  filled  with  mercury.  When  the 
gate  valves  are  opened,  the  pressure  of  the  water  causes  the 
mercury  to  rise  or  fall  to  heights  which  indicate  the  pressures 
at  the  points  of  the  main  to  which  the  gauge  is  attached.  A 
graduated  scale  allows  a  comparison  of  the  pressures. 


PRESSURE  FROM 
LOWER  END  OF. 
VENTURI  METER 


PRESSURE  FROM 

THROAT  OF 
VENTURI  METER 


FIG.  18. 

The  difference  between  the  pressures  at  the  full  section  and 
the  section  most  contracted  indicates  the  difference  between 
the  velocities  at  the  two  points;  the  difference  between  the 
pressures  at  the  full  sections  above  and  below  the  contraction 
corresponds  to  the  loss  of  head  between  the  two  points. 

Denoting  the  area  of  the  section  not  contracted  by  A,  the 
area  of  the  section  most  contracted  by  a,  and  the  difference 
between  the  pressures  converted  into  feet  of  head  of  water  by 
H,  the  theoretical  quantity  passing  through  the  section  most 
contracted  per  second  is  given  by  the  equation 

A  —  a 


For  the  actual  discharge  this  is  multiplied  by  a  coefficient, 
which,  however,  differs  little  from  unity.    In  the  Venturi  meter, 


DISCHARGE   OF   CONDUITS   UNDER  PRESSURE  187 

as  usually  constructed,  the  area  of  the  throat  is  contracted  to 
one  ninth  the  area  of  the  full  section  of  the  main.  Its  length  is 
from  eight  to  sixteen  times  the  diameter  of  the  full  section.  It 
has  a  registering  device  which  mechanically  converts  differences 
of  pressure  into  corresponding  velocities  and  these,  for  a  given 
diameter  and  a  certain  interval  of  time  (10  minutes),  into 
gallons  of  discharge. 

The  meter  is  made  in  sizes  from  2  up  to  100  inches  in  diameter. 
The  loss  of  head  is  insignificant  and  the  condition  of  the  water 
does  not  affect  its  working. 

The  discharge  from  vertical  tubes  was  recently  determined 
by  Lawrence  and  Braunworth  and  formulae  deduced,  which  not 
only  will  prove  to  be  of  great  value  in  computing  the  discharge 
of  artesian  wells,  but  furnish  another  method  to  determine  the 
discharge  of  any  conduit  under  pressure  with  a  fair  degree 
of  accuracy.  To  do  this,  it  will  simply  be  necessary  to  give 
the  end  of  the  conduit  a  vertical  direction  and  observe  the 
elevation  of  the  crest  of  the  outflow  above  the  rim  of  the 
conduit. 

The  investigators  mentioned  experimented  with  tubes  rang- 
ing between  2  and  12  inches  in  diameter  and  15  feet  long  and 
three  conditions  of  out-flow  were  observed,  depending  on  the 
pressure  head.  Under  a  feeble  head  the  water  flows  simply 
over  the  rim  of  the  conduit  as  it  does  over  a  sharp  edged  weir 
and  the  discharge  is  equal  to 

Q  -  8.8  d1'25  hlM. 

When  the  issuing  water  forms  a  jet  the  discharge  is  equal  to 
Q  =  5.57^'"  ft0'53, 

in  which  Q  =  cub.  ft.  per  sec. 

d  =  actual  internal  diameter  in  feet. 

h  =  elevation  of  crest  of  water  above  the  rim  of  the 
conduit,  in  feet,  determined  by  sighting  rod.  For  the  condition 
intermediate  between  the  weirflow  and  the  jetflow  no  formula 
was  deduced. 


188  THE   FLOW   OF   WATER 

Discharge  of  Open  Conduits. 

c. 

When  the  discharge  of  an  open  conduit  cannot  be  measured 
by  a  weir  or  an  orifice,  it  is  necessary  to  find  the  mean  velocity 
of  flow. 

The  mean  velocity  in  a  vertical  section  is  ascertained  directly 
by  means  of  rod-floats  or  by  making  measurements  at  the  point 
where  the  thread  of  mean  velocity  is  found,  either  with  a  current 
meter  or  with  a  double  float.  Indirectly  the  mean  velocity  is 
found  by  means  of  surface  floats  or  by  current  meter  observa- 
tions at  different  points  in  the  vertical  section. 

If  the  channel  is  narrow,  measurements  in  one  vertical  section 
are  generally  sufficient,  especially  if  a  rod-float  is  used.  With 
increasing  width  of  the  channel  observations  in  two  or  more 
vertical  sections  are  necessary. 

When  the  mean  velocity  of  flow  in  a  river  is  to  be  ascertained, 
the  channel  is  divided,  at  right  angles  to  the  line  of  flow,  into 
sections  5,  10,  20  or  more  feet  wide,  the  distance  depending  on 
the  degree  of  accuracy  desired. 

The  mean  velocity  at  each  section  is  found  by  means  of  rod- 
floats,  by  observations  at  the  surface,  at  mid-depth,  at  the 
position  of  the  thread  of  mean  velocity  or  at  points  of  propor- 
tional depth.  The  mean  velocity  for  the  whole  channel  is  found 
by  taking  the  mean  of  the  mean  velocities  of  all  the  sections. 

For  the  discharge  of  the  whole  channel  the  mean  velocity  of 
each  section  is  multiplied  by  its  area  and  the  discharges  of  all 
the  sections  summed  up.  If  floats  are  used,  the  stretch  over 
which  the  float  is  to  pass  should  be  carefully  measured  and 
staked  off.  If  possible  ropes  or  wires  should  be  stretched  across 
the  stream,  at  right  angles  to  the  line  of  flow.  The  float  should 
be  started  some  distance  above  the  rope  and  the  time  of  its 
passage  carefully  observed. 

The  distance  measured  out  may  be  250  to  500  feet  for  swift 
streams;  50  feet  will  suffice  if  the  current  is  feeble.  The  longer 
the  stretch  the  more  reliable  the  time  observation.  On  the 
other  hand,  if  the  stretch  is  long  it  is  often  exceedingly  difficult 


DISCHARGE   OF  OPEN   CONDUITS 


189 


to  keep  the  float  in  a  position  parallel  to  the  axis  of  the  stream. 
This  is  especially  so  near  the  banks.  On  this  account  it  may 
be  necessary  to  measure  stretches  as  short  as  20  feet. 

A  surface  float  may  be  a  ball  of  wood  or  some  other  light 
material,  or  else  a  watertight  metal  cylinder,  so  loaded  as  to 
float  flush  with  the  surface  of  the  stream.  A  small  flag  will 
render  the  float  more  visible. 

Double  floats  are  used  to  find  the  velocities  at  different  depths 
below  the  surface.  They  consist  of  light  surface  floats  con- 
nected by  a  fine  strong  cord,  to  a  large  sub-surface  float.  A 
ball  of  wood  or  a  flat  watertight  metal  box  makes  a  good  surface 
float,  a  watertight  metal  cylinder,  heavy  enough  to  keep  the 
cord  in  tension,  but  not  to  drag  it  below  the  surface  is  an  excellent 
sub-surface  float.  The  speed  of  the  surface  float  is  identical 
with  that  of  the  larger  float  and  observations  of  its  passage  will 
give  the  speed  of  the  latter. 

Usually  the  subsurface  float  is  placed  at  the  point  where  the 
thread  of  mean  velocity  is  found.  The  use  of  double  floats 
generally  leads  to  trouble  of  one  kind  or  another;  they  are  rarely 
used,  except  to  measure  velocities  in  very  deep  channels. 

A  cylindrical  wooden  pole  two  inches  in  diameter  and  loaded 
at  the  bottom,  so  that  it  will  float  vertically,  makes  an  excellent 
rod-float.  It  may  be  made  in  sections  and  screwed  together. 
A  brass  cylinder  screwed  to  the  bottom  makes  an  excellent 
weight.  Into  it  shot  may  be  placed  to  suit  the  weight  to  all 
requirements.  Watertight  tin  tubes  also  make  good  rod- 


FIG.  19.    Channel  of  River  Divided  into  Vertical  Sections. 

floats.  Rod  floats  should  be  loaded  so  that  they  nearly  reach 
to  the  bottom  of  the  channel,  but  never  touch  it.  On  the  other 
hand  they  must  not  be  too  short,  or  else  they  will  travel  with 
a  speed  exceeding  the  mean  velocity. 


190 


THE    FLOW    OF   WATER 


The  rod-float  is  the  ideal  instrument  to  measure  the  velocity 
in  a  flume  or  aqueduct.  The  fact  that  it  integrates  the  velocity 
of  the  whole  section  and  thus  indicates  the  mean  velocity  directly 
is  an  advantage  not  possessed  by  any  other  measuring  device. 
If  properly  used  it  gives  results  whose  accuracy  cannot  be  ques- 
tioned. However,  if  the  bottom  of  the  channel  is  very  rough, 
covered  with  plants  or  else  very  deep,  its  use  is  not  indicated. 


FIG.  20. 

Velocity  measurements  are  made  in  the  centre  of  each  sec- 
tion. Depths  are  taken  by  soundings. 

Line  (1)  indicates  the  position  of  the  thread  of  maximum 
velocity  in  each  section,  line  (2)  the  position  of  the  thread  of 
mean  velocity  in  each  section,  and  line  (3)  the  position  of  the 
thread  of  mean  velocity  for  the  whole  section. 

The  current  meter,  like  so  many  other  hydraulic  measuring 
devices,  originated  centuries  ago  in  the  Valley  of  the  Po,  Italy, 
the  cradle  of  hydraulics.  The  earliest  form  consisted  of  a  small 
paddle  wheel  mounted  in  a  floating  frame.  It  could  only  be 
used  at  the  surface. 

When  Woltman,  in  1790,  added  a  recording  device  the  instru- 
ment could  be  used  at  any  depth.  The  recording  mechanism 
consisted  of  an  endless  screw  fitted  to  the  horizontal  axis;  and  a 
series  of  toothed  wheels  which  transmitted  the  motion  of  the 
axis  to  a  register.  The  recording  mechanism  was  thrown  in  and 
out  of  gear  by  a  string,  attached  to  a  lever.  The  instrument 
was  fitted  and  clamped  to  a  one-inch  pole  on  which  it  could  be 


DISCHARGE  OF  OPEN  CONDUITS  191 

slid  up  and  down.  To  read  the  number  of  revolutions  recorded 
the  current  meter  had  to  be  taken  out  of  the  water.  The  instru- 
ment was  generally  known  as  "Wolt man's  Tachometer." 

Many  modifications  of  this  instrument  appeared,  mostly  of 
the  windmill  pattern,  with  propellers  and  vanes.  Some  have 
the  axis  of  the  propeller  horizontal,  others  vertical,  and  the 
shape  of  the  propellers  is  variable.  The  general  form  of  the 
instrument  is,  however,  always  the  same.  The  present  day 
current  meter  has  an  electrical  signalling  or  registering  device. 
The  best  known  patterns  are  those  of  Harlacher  in  Europe,  and 
those  of  Price  and  Ritchie-Haskell  in  the  United  States. 

The  Harlacher  meter  is  of  the  windmill  pattern ;  its  propeller 
has  four  blades.  A  vane  about  12  inches  long  and  5  inches  wide 
is  fitted  to  a  prolongation  of  the  axis  of  the  wheel.  This  direct- 
ing device  keeps  the  face  of  the  wheel  at  right  angles  to  the  line 
of  flow.  To  the  axis  of  the  propeller  is  fitted  an  endless  screw, 
operating  a  toothed  wheel.  A  pin  in  the  side  of  the  wheel 
strikes  an  electric  wire  at  each  revolution,  thus  completing  an 
electrical  circuit.  The  battery  with  the  registering  or  sounding 
device  is  kept  at  the  surface.  The  meter  slides  up  or  down  on 
a  vertical  rod.  To  move  the  meter  up  and  down  with  a  uniform 
speed  an  apparatus  consisting  of  ropes,  pulleys  and  weights  is 
often  used. 

The  propeller  of  the  Price  current  meter  has  four  cup-shaped 
wings;  its  axis  is  vertical  and  its  revolutions  are  indicated  by  an 
electrical  buzzer.  The  instrument  is  generally  used  without  a 
rod;  it  is  kept  vertical  by  a  weight  attached  to  the  frame  and 
moved  up  and  down  by  a  cord.  Its  vane  consists  of  two  blades, 
one  horizontal,  the  other  vertical,  intersecting  in  the  middle 
at  right  angles.  It  is  made  in  two  sizes.  The  small  meter 
measures  velocities  as  low  as  0.2  feet  per  second  with  a  fair  degree 
of  accuracy;  the  large  meter  gives  good  results  down  to  velocities 
of  0.5  foot  per  second. 

The  latest  design  in  the  line  of  meters  is  the  Ritchie-Haskell 
so-called  "direction  current  meter."  Like  the  Harlacher  and 
the  Price  this  instrument  has  a  device  recording  the  number  of 
revolutions  of  the  propeller  electrically.  It  has  also  a  device 


192  THE  FLOW  OF   WATER 

indicating  the  direction  of  the  current.  The  body  of  the  instru- 
ment is  a  compass  with  a  magnetic  needle.  An  electrical  circuit 
measures  the  angle  between  the  direction  of  the  needle  and  the 
direction  in  which  the  vane  points  and  indicates  the  angle  on  a 
graduated  dial. 

Current  meters  must  be  rated;  that  is,  the  relation  between 
the  velocities  and  the  number  of  revolutions  of  the  propeller 
must  be  ascertained.  This  is  done  by  pulling  the  meter  at 
various  constant  speeds  through  a  still  body  of  water,  and  deter- 
mining the  relation  between  speeds  and  revolutions. 

Current  meters  as  furnished  by  the  makers  are  always  rated, 
but  they  must  subsequently  be  rerated  at  frequent  intervals, 
if  good  results  are  desired.  As  with  floats,  measurements  with 
the  current  meter  are  made  in  various  ways.  The  best  method 
is  no  doubt  the  one  adopted  by  Harlacher  of  sliding  the  instru- 
ment by  means  of  a  mechanism  at  a  uniform  speed  up  and  down 
on  a  pole.  By  this  process  the  velocity  of  the  whole  section  is 
integrated  and  a  very  good  mean  value  found.  If  no  pole  is 
used  the  instrument  is  most  conveniently  moved  up  or  down  by 
means  of  a  cord  thrown  over  a  small  pulley. 

A  good  current  meter,  properly  rated  and  carefully  handled, 
surpasses  any  other  instrument  in  the  facility  and  extent  of  its 
application;  it  gives  results  nearly  as  trustworthy  as  the  rod- 
float,  and  for  average  velocities  nearly  as  accurate  as  a  weir. 

The  Darcy  gauge,  an  instrument  formerly  in  great  favor,  is  at 
present,  owing  to  the  great  perfection  of  the  current  meter, 
but  rarely  used.  The  instrument  consists  of  a  combination  of 
two  Pitot  tubes,  fastened  to  a  supporting  frame. 

A  Pitot  tube  is  a  vertical  glass  tube  with  a  right-angled  bend. 
If  such  a  tube  is  placed  into  a  stream,  with  its  mouth  facing  up- 
stream and  at  right  angles  to  the  line  of  motion,  the  water  will 
ascend  in  the  tube  to  a  height  which  is  equal  to 

v2 
b=*—,  nearly. 

If  the  mouth  of  the  tube  faces  the  bank  of  the  stream,  and  is 
in  line  with  the  line  of  motion,  there  will  be  no  difference  of 
level  between  the  surface  of  the  water  in  the  tube  and  the 
surface  of  the  stream. 


SURFACE   MEAN  AND  BOTTOM  VELOCITIES  193 

If  the  mouth  of  the  tube  faces  downstream  and  is  at  right 
angles  to  the  line  of  motion,  the  surface  of  the  water  in  the  tube 
will  be  below  the  surface  of  the  stream,  the  difference  being 
equal  to 


In  this  case  the  velocity  is  somewhat  modified  by  the  retard- 
ing influence  of  the  tube.  Darcy  combined  two  tubes  having 
their  mouths  at  right  angles,  and  provided  their  lower  parts 
with  stopcocks,  which  can  be  operated,  when  the  instrument  is 
in  the  water,  by  means  of  a  string.  If  the  cocks  are  open  and 
the  mouth  of  one  of  the  tubes  faces  upstream  at  right  angles  to 
the  line  of  motion  the  water  will  ascend  in  it  while  it  will  not 
ascend  in  the  other  tube.  If  the  corks  are  then  closed,  the 
instrument  may  be  lifted  out  of  the  water  and  the  difference  of 
level  in  the  two  tubes  read  off  on  a  graduated  scale. 

Surface  Mean  and  Bottom  Velocities. 
Position  of  Thread  of  Mean   Velocity. 

From  82  observations  of  flow  in  small  channels  Bazin  deduces 
the  following: 

Mean       Velocity  =  Maximum   Velocity  —  25.4 
Bottom   Velocity  =  Maximum  Velocity  —  36.3    Vr.s 
Bottom   Velocity  =  Mean          Velocity  —  10  .  87  \/r\s 
From  this  we  have 

Fmean  +  25.4  Vi\s 


V  max. 
V  mean  1 


1.0, 


V  max.      1  +  25.4  Vr.a 


and  as  — — •  =  c, 

r.s 


,  1  V  mean  1 

we  have  also  — = .,   .» 

V  max.       -       25.4 

c 

i  vi     ••  y  bottom  1 

and  likewise 


V  max.          1    ,  36.3 

1  H 

C 


194  THE  FLOW  OF  WATER 

Comparison  of  values  of  __  mean  =  — —  with  values  of 

V  max.       1      25.4 

c 

mean  founcj  ^  observations  of  flow  in  a  great  variety  of 
V  max. 

channels  shows  that  Bazin's  formula  is  not  of  general  application. 
It  fails  because  the  influence  of  the  value  of  the  total  depth  of 
the  channel  is  not  considered. 

V  mean 
The  following  values  of  — —  are  given  by  the  most  reliable 

authorities : 

V  mean 
V  surface 

Revy,  Parana  de  las  Palmas,  La  Plata 0.835 

Harlacher,  Bohemian  Rivers,  28  observations 0.838 

Swiss  Engineers,  Swiss  Rivers,  200  observations 0.835 

Lippincott,  Sacramento  River,  Cal.,  Depth,  3-5  feet  ...  0.88 
Lippincott,  Tuolumne  River,  Cal.,  Depth,  1.12-1.84  feet  .  0.88 
Lippincott,  San  Gabriel  and  Santa  Anna,  Rough  channels, 

10-20  feet  wide.     Depth,  0.25-1.0  feet     0.92 

Pressey,  Catskill  Creek,  partial  section 0.82 

Pressey,  Fishkill  Creek,  partial  section      0.93 

Pressey,   Mean  of  28  observations  of  flow  in  rivers  with 

rough  bottoms,  Average  depth,  5.05  feet 0.80 

Prony,  Small  wooden  channels 0.8164 

Prony  and  Destrem,  Neva  River,  Russia 0.78 

Boileau,  Canals 0.82 

Baumgartner,  Garrone  River,  France 0.80 

Cunningham,  Solani  Aqueduct 0.823 

Humphreys  &  Abbot,  Mississippi 0.79-0.82 

From  these  and  other  data  given  by  Murphy  (Cornell  testing 
flume)  and  others,  the  writer  found  that  the  relation  between 
the  surface  velocity  and  the  mean  velocity  may  be  expressed  by 
the  equation 

Mean  velocity  = j-=.  surface  velocity          (1) 

!+»-£ 

Vc 

in  which  n  is  a  coefficient  ranging  in  value  between  0.25  for  the 
roughest  and  0.35  for  the  smoothest  classes  of  conduits. 

Its  value  is 

n  =  0.32  for  K  =  1.25 
n  =  0.30  for  K  =  1.75 
n  =  0.27  for  K  =  2.25 

For  the  velocity  at  any  point  x,  depth  d,  in  the  vertical  section 


SURFACE  MEAN  AND  BOTTOM  VELOCITIES 


195 


we  found  from  data  relating  to  flow  in  channels  with  rough 
bottoms,  such  as  rivers  with  detritus  or  coarse  gravel, 

1 


1  + 


§)' 


(2) 


in  which  D  is  the  total  depth.  This  is  on  the  assumption  that 
the  bottom  velocity  is  equal  to  one  half  the  surface  velocity,  a 
relation  which  holds  good  only  for  channels  with  rough  bottoms. 
Bazin  found  from  observations  of  flow  in  small  artificial  channels 
that  the  difference  between  the  surface  and  the  bottom  velocity 
ranges  between  0.25  and  0.5  of  the  surface  velocity,  the  differ- 
ence increasing  in  value  with  'the  roughness  of  the  walls.  In 
canals  and  rivers  with  comparatively  smooth  bottoms  the 
difference  ranges  between  0.3  and  0.4,  the  average  difference 
being  0.35  of  the  surface  velocity. 

Combining  the  two  equations  (1)  and  (2),  we  have  for  the 
position  of  the  thread  of  mean  velocity  in  the  vertical  section 
of  rivers  and  canals  with  somewhat  rough  bottoms  and  whose 
width  is  several  times  the  depth 


41) 


(3) 


as  the  depth  below  the  surface  at  which  the  thread  of  mean 
velocity  is  found.  The  formula  does  not  apply  to  flumes  and 
other  narrow,  deep  channels. 

From  equations  (1)  and  (3)  we  find  the  following  values  of 
mean 


the  relation 


relative  position  of  the  thread 


V  surface 

of  mean  velocity  in  a  vertical  section,  assuming  K  =  1  .  0  and 
v  =  3  feet. 


V  mean 

Relative 

E> 

V  mean 

Relative 

V  surface 

Depth. 

V  surface 

Depth. 

1.0 

0.898 

0.538 

10.0 

0.854 

0.604 

2.5 

0.881 

0.563 

15.0 

0.842 

0.616 

5.0 

0.369 

0.583 

25.0 

0.832 

0.631 

7.5 

0.859 

0.596 

30.0 

0.813 

0.656 

APPENDIX  I. 


Variation  of  the  Coefficient  c  with  the  Slope. 

IN  the  preceding  chapters  we  have  defined  the  variation  of 
the  coefficient  c  with  the  mean  hydraulic  radius,  with  the  degree 
of  roughness  of  the  wet  perimeter  and  with  the  velocity  of  flow. 
We  will  now  proceed  to  investigate  if  it  is  possible  to  find  a 
true  expression  for  the  variation  of  the  coefficient  c  with  the 
slope  by  the  graphical  method.  From  Formula  III  we  have 
66  (  *f?  +  m)  V*  =  c, 


66  (       +  m) 

I          ^          \9 
v  =  [  -  1  • 

\66  (S/r  +  m)/' 
or  substituting  for  v  its  equivalent 

(66  ($fr  +  m)  Vr7s)*  =  (  -  -4  -  Y; 

V66  (  Vr  +  m)/ 

hence  66  (  tfr  +  m)  Vr  .  s    =  (  -  T=  -  )  , 

\66  (  -N/F  +  m)) 


_ 

and  (66  (  Vr  +  m))9  Vr  .  s  =  c8; 

consequently 

(66(Vr+m))*(r.a)*-  c; 

or  66  (  Vr  +m)  (66  (  Vr  +  m))*  (r  .  s}&  =  c. 

This  goes  to  show  that  c  increases  with  (rs)^-,  consequently  the 
variation  of  the  coefficient  c  with  the  slope  depends  on  the 
value  of  R. 

The  variation  of  the  coefficient  c  follows  the  law  of  the  para- 
bola. If  values  of  the  coefficients  a  =  V%  and  V&  are  plotted  as 
ordinates  to  values  of  v  as  abscissae,  the  points  so  found  lie  in 
curves  which  are  parabolas  of  the  ninth  or  eighteenth  order.  A 
curve  somewhat  resembling  a  parabola  is  the  equilateral  hyper- 

196 


APPENDIX   I 


197 


bola,  and  it  is  possible  to  draw  a  curve  of  this  kind  which  nearly 
coincides  with  the  parabola. 

The  equation  of  the  equilateral  hyperbola  concave  towards 
the  axis  of  abscissae  may  be  put  into  the  simple  form 


The  curve  in  Fig.  1  represents  the  hyperbola  of  this  equation. 
In  the  figure  ZO  is  the  vertical  asymptote, 
Zd  the  horizontal  asymptote, 
YK  the  axis  of  ordinates, 
KX  the  axis  of  abscissae, 
Zg    the  axis  of  the  hyperbola, 
X    the  distance  between  the  vertical  asymptote 

ZO  and  the  axis  of  ordinates  YK, 
c    the  ordinate  of  any  point  in  the  curve. 


The  area  of  the  rectangle  ZOKY  is  the  constant  which  deter- 
mines the  hyperbola.  It  is  equal  to  the  square  zfgh  or  the  area 
of  any  rectangle  comprised  between  the  asymptotes  and  per- 
pendiculars drawn  to  them  from  any  point  in  the  curve. 


198  THE   FLOW   OF   WATER 

Consequently,  if  lines  are  drawn  from  the  center  z  to  points  R 
on  the  axis  of  abscissae,  these  lines  will  intersect  the  axis  of 
ordinates  in  points  which  give  the  values  of  c  corresponding  to 
the  values  of  R.  In  this  way  the  hyperbola  may  be  easily 
constructed. 

Bazin  in  his  paper,  "  Etude  d'une  nouvelle  formule,"  etc.,  put 
the  equation  for  the  coefficient  c  into  the  form 


in  which  y  is  constant  and  equal  to  157.5  in  English  measure, 
and  0,  a  variable,  indicating  the  degree  of  roughness. 
Dividing  by  y  we  have 

c  =  • substituting  x  for  g. 

y      yVr 

Transposing  we  have 

11      xj_ 

c      y      y  \/r 

This  is  the  equation  of  a  straight  line  having  values  of  — _  as 

Vr 

abscissae,  values  of  -  as  ordinates.     If  this  equation  would  hold 
good,  points  of  values  of  -  pertaining  to  one  slope  would  lie  in 

Of 

straight  lines  intersecting  the  axis  of  ordinates  in  a  point—.     If, 

y 

however,  values  of  -  and  — —  are  plotted  as  indicated  it  appears 

c          vr 

that  only  those  points  -  pertaining  to  data  of  flow  in  old  pipes 

C 

or  fairly  regular  channels  in  earth  lie  in  straight  lines,  while 
those  pertaining  to  data  of  flow  in  very  smooth  conduits  lie  in 


APPENDIX   I 


199 


curved  lines  convex  towards  the  axis  of  abscissae,  and  those 
pertaining  to  data  of  flow  in  very  irregular  channels  lie  in  curved 
lines  concave  towards  the  axis  of  abscissae. 

If  straight  lines  are  drawn  averaging  between  the  points  as 
much  as  possible,  these  lines  will  intersect  the  axis  of  ordinates 


FIG.  2. 


in  points  giving  values  of  -  for  the  greatest  value  of  v  and  the 

y 

greatest  value  of  R  included  in  the  series  plotted.  These  lines 
will  also  intersect  the  axis  of  abscissae  in  points  which  give  the 

value  of  -  pertaining  to  each  value  of  - .    In  Fig.  2  we  thus  plotted 

the  experimental  data  of  Darcy-Bazin,  series  7,  8,  9  and  one 
series  given  by  Rittinger  (s  =  0.0343),  all  pertaining  to  flow  in 
testing  channels  of  rough  boards. 

It  will  be  observed  that  the  lines  pertaining  to  the  steeper 

slopes  intersect  each  other  in  a  point  whose  abscissa  for  —  is 

Vr 

1.0.  This  is  due  to  the  fact  that  for  the  greater  slopes  s  =  0.0049, 
0.00824,  and  0.0343,  the  velocity  is  so  high  that  c  varies  but 
very  little,  while  it  varies  much  for  the  feebler  slope  s  =  0.0015. 


200  THE   FLOW  OF  WATER 

The  highest  value  of  -  corresponding  to  —  -  =  1.0  is  0.0084,  the 
c  Vr 

lowest  0.0080,  average  0.0082.  Denoting  the  abscissa  of  the 
point  of  intersection  by  a  and  the  average  ordinate  by  K  we 
have 


y    y  a 

and  x  =  Kay  —  a, 

TZ 

and  considering  —  =  Ka  as  a  tangent  and  denoting  it  by  I, 

a 

we  have  x  =  ly  —  a. 
Consequently  in  our  case 

x  =  0.0082  y  -  1.0, 

which  gives  values  of  x  very  nearly  equal  to  those  found  graph- 

ically.   This  formula  will,  however,  only  hold  good   for  the 

values  of  R  included  in  the  series,  the  highest  of  which  is  1.0. 

In  Fig.  3  the  values  of  y  found  graphically  from  Fig.  2  are 

plotted  as  ordinates  to  values  of  -  as  abscissae.     The  points  y 

s 

are  seen  to  lie  in  a  curved  line,  intersecting  the  axis  of  ordinates 
at  a  point  B  =  131.0  nearly.  If  the  line  CD  is  produced,  it 
will  intersect  the  axis  of  ordinates  in  y  =  157.5,  which  is  the 
constant  in  the  formula  of  Bazin  mentioned  above.  The  tan- 
gent of  the  angle  CEF  (in  this  instance  0.29)  corresponds  to 
Bazin's  coefficient,  g,  indicating  the  degree  of  roughness.  The 
value  of  m  obtained  from  the  given  data  is  0  .  70  ;  hence  the  value 
of  ^r  +  m,  for  the  highest  value  of  R  is  1  .70.  Dividing  131  .0 
by  1  .  70  we  have 

y'  =  77  (  'N/r  +  m),  nearly,. 

as  the  value  of  y  corresponding  to  the  highest  velocity  included 
in  the  series  plotted. 

If  from  the  point  B  =  131  .  0  =  77  (  Vr  +  m)  a  line  is  drawn 
parallel  to  the  axis  of  abscissae,  any  increase  in  the  value  of 


APPENDIX  I 


201 


77  ( 'N/r  +  m)  due  to  any  slope  less  than  0.0343  will  appear  as 
an  ordinate  above  this  line  BG. 

It  will  be  observed  that  values  of  y"t  y'"y  s/v,  etc.,  increase 

with  the  decrease  of  the  slope  or  increasing  values  of  -  •    We 


nay  therefore  put  y"  ',  y'",  etc.,  =  77  (  Vr  +  m)  +  -  in  which 

o 

j  is  a  coefficient  still  to  be  determined. 

C 

^-^^^ 

C 
F 

B^^^ 

.  —  "* 

G 

" 

yll 

- 

,» 

Values  of  IT 

FIG.  3. 

The  line  RDC  is  evidently  a  parabola.  If  a  line  is  drawn 
from  B  to  C  the  tangent  of  the  angle  CBG  will  be  equal  to  z, 
for  s  =  0 . 0015.  For  this  slope  we  have  from  the  figure 

y  =  198.0 
2/'  =  131.0; 

therefore         -  =  198  -  131  =  67  and  z  =  0.0015  X  67, 

s 

or  z  =  0.1005. 

Hence  y  =  77  (  N/r  +  m)  +  -1—  • 

s 

From  experimental  data  pertaining  to  flow  in  small  channels 
in  earth,  R  ranging  between  1  and  1 . 75  (Darcy-Bazin,  Grosbois 


202  THE   FLOW  OF   WATER 

canal)    which   are,    however,    somewhat   doubtful,    we    found 
z  =  0.  0936,  while  from  data  pertaining  to  flow  in  the  La  Plata 
and   its  tributaries  we  found  0  =  0.00293.     From  this  it  is 
evident  that  z  is  a  variable  and  that  its  value  depends  on  the 
value  of  R.    Having  found  an  expression  for  y,  the  value  of  x 
may  be  found  from  experimental  data  without  resorting  to 
graphical  methods. 
No.  12,  series  6,  Darcy-Bazin  gives 
R  =  0.922 
s  =  0.00208 
c  =  118.9. 

Hence  y  =  77  (  ^922  +  0.68)  + 


or  y  =  127.87  +  47.6  =  175.47. 

Dividing  175.47  by  c  =  118.9   we  find  x  =  1.475, 

=  1  +0.475.    But  0.475  is  equal  to  0.01,      **'*     or  0.01,  47.6. 

U  . 


Denoting  the  term  0.01,  which  is  variable,  by  I,  we  have  from 
the  given  data  for  the  variation  of  the  coefficient  c  with  the 
slope  the  expression 

77  (tyr  +  m)+— L— 

s 
c  = :r~: ; ' 


which,  within  certain  limits  corresponds  to 

c  =  66  (  S/r  +  m)  V&. 

From  data  relating  to  flow  in  a  semicircular  channel  of  rough 
boards  (Darcy-Bazin,  series  26)  we  find 

01  * 

y  +  —  =  210,  hence  y  =  210  -  67, 

s 

which  is  equal  to 

0.1 


84  (  Vr  +  m)  + 


APPENDIX  I  203 

and  which  corresponds  within  certain  limits  to 
c  =  66  ( Vr  +  m)  F^. 

Dividing  84  by  66  the  quotient  1.272  is  the  value  of  the 
coefficient  of  variation  of  c  for  v  =  18.0.  Hence  v  =  18.0  is 
the  limit  up  to  which  the  formula  holds  good. 

The  formula  apparently  gives  good  values  of  c  up  to  the 
limit  indicated.  By  trial  we  find,  however,  that  it  does  not 
hold  for  values  of  R  greater  than  1.0;  unless  rs  is  substituted  in 
the  equation  for  s.  Consequently  the  variation  of  c  with  the 
slope  is  dependent  on  the  value  of  R,  a  fact  we  demonstrated  at 
the  beginning  of  this  chapter. 

The  facts  related  plainly  show  that  a  formula  derived  in  the 
manner  indicated  can  only  be  of  limited  application.  It  holds 
good  only  within  the  range  of  values  of  R,  s  and  m  included 
in  the  series  of  data  from  which  it  is  derived.  In  other  words: 
We  cannot  get  out  of  a  formula  what  we  do  not  put  into  it. 

A  general  formula,  like  that  of  Ganguillet  and  Kutter,  derived 
by  the  methods  we  have  indicated,  cannot  embody  true  laws  of 
flow,  it  naturally  must  be  deficient  in  one  respect  or  the  other. 
The  more  so,  if  the  data  on  which  the  formula  is  based  are 
erroneous.  The  experimental  data  derived  from  observations 
of  flow  in  the  lower  Mississippi  by  Humphreys  and  Abbot  and 
embodied  by  Ganguillet  and  Kutter  in  their  formula  have  been 
found  to  be  incorrect,  greatly  at  variance  with  those  time  and 
again  found  by  the  United  States  engineers.  The  contention  of 

Ganguillet  and  Kutter,  that,  if  values  of  -  are  plotted  as  ordinates 

c 

to  values  of  — -  as  abscissae  and  lines  drawn  through  all  the  points 

Vr 

-  these  lines  will  intersect  each  other  in  a  point  —  =  1  meter, 

c  Vr 

and  that  therefore  c  will  increase  with  increasing  values  of  s  if 
R  is  less  than  1  meter,  and  decrease  if  R  is  greater  than  1  meter, 
is  also  plainly  a  fallacy. 

If  values  of  —  and  — -  derived  from  the  numerous  series  given 


204  THE  FLOW  OF   WATER 

by  Darcy-Bazin  are  plotted  as  indicated,  it  will   be  observed 

that  for  many  of  the  series  the  lines  intersect  at  —  =  1  foot. 

Vr 

It  would  be  absurd,  however,  to  draw  the  conclusion  there- 
from that  c  will  increase  or  decrease  with  increase  of  the  slope 
if  R  is  less  or  greater  than  1  foot.  The  intersection  of  the  lines 

at   —  =  1  foot  is  due  to  the  fact,  that  for  the  greater  slopes 

values  of  c  are  nearly  constant  for  values  of  R  equal  for  1  foot  or 
more,  because  the  value  of  F1*  increases  slowly  at  high  velocities. 


APPENDIX   II. 

The  Formula  in  Metric  Measure. 

THE  general   equation    for  the  velocity  of   flow  reads,  for 
Metric  measure, 

V  =  50(-N/r>m)  VrTs 

2gH 

0.007844     L 


^Jr  +  m)2  R_ 

The  coefficients  of  variation  of  c  are  equal,  as  for  English 
measure,  to 


a  =  V^  holds  good  also  for  semicircular  open  conduits. 

Values  of  the  coefficient  m,  indicating  the  degree  of  rough- 
ness, are  found  in  the  following  table,  mE  signifying  the  English 
and  mM  the  Metric  values. 


Exponential  Equations. 

The  constants  of  the  exponential  equations  which  we  have 
found  for  English  measure  are  converted  into  Metric  equivalents 
'by  putting 

log  constant  Metric  measure  =  log  constant  English  measure 

+   "    3.281s 
-   "    3.281. 

x  being  the  variable  power  of  R  or  D. 

The  equations  for  conduits  under  pressure  are  as  follows, 
diameters  being  in  meters,  velocities  in  meters  per  second  and 
quantities  in  cubic  meters  (1000  liters)  per  second : 

205 


206 


THE   FLOW   OF   WATER 


VALUES  OF  mE  WHICH  APPLY  IN  THE  ENGLISH  AND  VALUES  OF  mM 
WHICH  APPLY  IN  THE  METRIC  SYSTEM. 


mE 


mM 


Description  of  Conduits. 


1.0 
0.95 

0.85 
0.83 

0.80 
0.70 

0.68 


0.57 

0.53 
0.50 
0.45 

0.30 

0.20 

0 

0.10 

0.20 

0.27 

0.32 


0.85 
0.80 

0.70 
0.75 

0.65 
0.62 

0.60 


0.48 

0.45 
0.47 
0.42 

0.25 
0.20 
0 

-0.1 
-0.2 
-0.27 
-0.32 


Semicircular  and  circular  conduits  lined  with  pure 

cement.     Long  straight  brass,  tin,  nickel  and  glass 

pipes. 
Rectangular  conduits  lined  with  pure  cement.     New 

pipes  of  planed  boards  and  very  smooth  asphalt- 
coated  cast  iron. 
Semicircular  conduits  lined  with  cement  plaster,  1  part 

cement,  2  parts  sand. 
Ordinary  new  straight  asphalt-coated  cast,  wrought 

iron  welded  and  wrought  iron  riveted  pipes  with 

screw  joints,   common   lead,  tin,  glass,  brass  and 

galvanized  pipes. 
Rectangular    conduits    lined    with    cement    plaster, 

smooth  concrete  or  very  good  brickwork. 
Semicircular  channels  lined  with  rough  boards .     Chan- 
nels   lined    with   fairly   good   brickwork   or   fairly 

smooth  concrete. 
Rectangular    channels     lined     with     rough    boards. 

Sewer  pipe  very  well  laid. 
Pipes    of    planed    boards,    asphalt-coated    cast    and 

wrought  iron,  riveted  wrought  iron  pipes  of  small 

diameters  or  with  screw  joints,  pipes  coated  with 

tar  or  lined  with  cement  or  smooth  concrete,  all 

some  time  in  use. 
Common  brickwork  or  concrete.     Very  good  ashlar 

masonry.     Ordinary  sewer  pipe. 

Asphalt-coated  riveted  pipe  above  3  feet  in  diameter. 
Channels  in  earth  roughly  lined  with  cement  mortar. 
Old  pipes  of  all  descriptions,  fairly  clean.  Channels 

lined  with  rough  brickwork  or  rough  concrete. 
Old  riveted  pipes  over  3  feet  in  diameter.     Ordinary 

ashlar  and  very  good  rubble  masonry. 
Channels    of    regular  cross-section  in  fine   cemented 

gravel.     Tile  drains. 
Channels  of  regular  cross-section  in  coarse  cemented 

gravel  or  rockwork. 
Channels  of  fairly  regular  cross-section  in  firm  sand 

or  sand  with  pebbles,  no  vegetation. 
Channels  in  earth  somewhat  above  the  average  in 

regularity  and  condition,  no  stones  or  vegetation. 
Ordinary  channels  in  earth,  with  stones  or  vegetation 

here  and  there. 
Channels  of  irregular  cross-sections  or  channels  of 

fairly    regular    cross-sections    but    with    stones    or 

plants. 
The  values  of  K  corresponding  to  in  =  —  0.1,  —  0.2, 

-0.27, -0.32  are  1.2,  1.5,  1.75,  1.93. 


APPENDIX   II 


207 


mE 

mM 

V  in 

Meters  per  Second. 

Q  in  Cubic  Meters  per  Second. 

0 

.95 

0 

83 

60. 

92 

£0.67 

S* 

47 

85 

£2-67 

S* 

0 

.83 

0 

75 

56. 

54 

£0.68 

« 

44 

41 

£2.68 

u 

0 

.68 

0 

60 

51. 

28 

£0-69 

II 

40 

28 

£2.69 

11 

0 

.57 

0 

48 

36. 

76 

£0.7 

s* 

28 

87 

£2.7 

at, 

0 

.57 

0 

48 

40. 

21 

£0.7 

Egg-shaped 

47 

40 

£2.7 

Egg-shaped 

0 

.53 

0 

45 

35. 

55 

£0.7 

«* 

27 

92 

£2.7 

stV 

0 

.45 

0 

26 

25. 

48 

£0-66 

S^ 

20 

0 

£2.66 

s^ 

0 

.30 

0 

22 

22. 

45 

£0.67 

u 

17 

64 

£2.67 

11 

Values  of  D0-'67,  etc.,  are  found  in  Table  E,  values  of  D2'67,  etc., 
in  Table  F. 

These  tables  give  the  values  of  the  powers  of  diameters  for 
diameters  of  0.05,  0.10,  0.15,  0.20,  0.25  meters,  etc.  These 
correspond  closely  to  2,  4,  6,  8,  10  inches,  one  foot  being  0.3048 
meters,  one  meter  39.4  inches.  In  order  that  the  powers  of  the 
diameters  found  in  Tables  E  and  F  may  apply  to  a  greater  range 
of  diameters  we  shall  find  equations  in  which  the  unit  is  1  deci- 
meter =  0.1  meter,  so  that  diameters  must  be  taken  in  deci- 
meters and  fractions  thereof.  The  results  will  be  velocities  in 
decimeters  per  second  and  quantities  in  cubic  decimeters  or 
liters  per  second. 

The  diameters  found  in  the  tables  as  0.05,  0.10,  0.15,  0.20, 
0.25,  0.50,  0.75  when  taken  as  fractions  of  a  decimeter  corre- 
spond to  0.2,  0.4,  0.6,  0.8,  1.0,  2.0,  3.0  inches  respectively. 

The  discharge  of  a  new  wrought  iron  pipe  (m  =  0.83)  of  one 
inch  diameter  (0.025  meter  or  0.25  decimeter)  for  a  slope  of 
1  : 100  is  for  instance : 
Q  =  92.78  (0.25)2'68  (0.01)  & 

=  92.78  .  0.02435  .  0.075  =  0.1694  liters  per  second  or  10.164 
liters  per  minute. 


mE 

mM 

Velocity  in  Decimeters  per 
Second. 

Discharge  in  Liters  per  Second. 

0.95 
0.83 
0.68 

0.57 
0.57 

0.53 

0.83 
0.75 
0.60 

0.48 
0.48 

0.45 

130.2     Z)0-67     ST* 
118.1     Z)0-68       " 
104.7     Z)0-69       " 

73.34  Z)0-7     S& 
80.23  Z)0-7  Egg-shaped 

70.93  Z)0-7     S& 

102.3     Z)2'67     S& 
92.78  Z)2'68       " 
82.23  Z)2-69       " 

57.60  D2'7      Sl9>7' 
94.58  Z)2-7    Egg-shaped 

55.71  Z)2'7     S& 

0.45 
0.30 

0.36 
0.22 

55.74  D°-66    S2 
48.0     D°-67       " 

43.78  Z)2-66     S* 
37.7     D2-67     " 

208 


THE   FLOW   OF   WATER 


Exponential  Equations  Relating  to  Flow  in  Open  Conduits. 

Of  the  following  sets  of  equations  the  first  three  relate  to  flow 
in  the  semicircle,  the  rest  to  flow  in  the  semisquare,  the  depth 
being  in  meters,  the  velocities  in  meters  per  second,  the  dis- 
charges in  cubic  meters  per  second : 


mE 

mM 

V  in  Meters  per  Second. 

Q  in  Cubic  Meters  per  Second. 

1.0 

0.85 

89.25  D°'68    S& 

70.12  Z)2-68     £" 

0.85 

0.70 

83.0     D0-69 

65.1     D2-69 

0.70 

0.62. 

74.0    D°-70 

58.2    D2-70 

0.95 

0.80 

73.3    D°-67    S 

r9r 

146.6    D2-67    S 

V 

0.80 

0.65 

67.6    D0-68 

135.2    D2-68 

0.70 

0.62 

64.0    D0-69 

128.0    D2-89 

0.57 

0.48 

59.0    Z>°-7 

118.0    D2'7 

0.50 

0.49 

57.0    D0-715 

114.0    D2'715 

0.45 

0.42 

54.7    D0-715 

109.4    D2'715 

0.30 

0.25 

49.1     D°-735 

98.2    D2'735 

0.20 

0.20 

44.6    Z>°-735 

89.2    D2-735 

0 

0 

29.2    D°'75    S* 

58.4    D2-75    S* 

K 

K 

1.2 

1.2 

26.75  Z>°-765     " 

53.5    JD2-765      " 

1.5 

1.5 

23  .  6     D°'775     " 

47.2    D2'775      " 

1.75 

1.25 

21  6    D°'785     " 

43.2    D2-785      " 

1.93 

1.93 

20.5     Z>°-795     " 

41.0    D2-795      " 

Of  the  following  equations  the  first  three  apply  to  any  depth 
of  water  in  the  semicircular  section,  the  rest  to  any  depth  of 
water  in  any  other  form  of  section,  R  being  in  meters,  velocities 
in  meters  per  second. 


mE 

mM 

Velocities  in  Meters  per 
Second. 

mE 

mM 

Velocities  in  Meters  per 
Second. 

1.0 

0.85 

142.4     E0'68     S& 

0.30 

0.25 

81.0     #°-735     S*7 

0.85 

0.70 

134.0     fl0'69 

0.20 

0.20 

74.0     .R0-735       " 

0.70 

0.62 

122.0     tf0.70 

1 

0 

0 

49         ft0'75      S* 

0.95 

0.80 

116        #°-67    ST7 

K 

K 

0.80 

0.65 

108        R0-68 

1.2 

1.2 

45.4     £°-785       " 

0.70 

0.62 

102.5     J?°'69 

1.5 

1.5 

40.4     .R0'775       " 

0.57 

0.48 

95.3     JR°'70 

1.75 

1.75 

37.2     #°-785       " 

0.50 

0.47 

92.3    .R0-715 

1.93 

1.93 

35.3     .R0-795       " 

0.45 

0.42 

89.0     #°-715 

APPENDIX  II  209 

English  and  Metric  Equivalents. 

The  following  relations  between  the  units  of  the  English  and 
the  Metric  Systems  of  Measurements  are  of  interest  in  their 
relation  to  the  flow  of  water. 

1  meter  =  10  decimeters  =  100  centimeters  =  1000  millimeters. 
1  sq.  meter  =  100  sq.  decimeters  =  10,000  sq.  centimeters. 
1  cu.  meter  =  10  hectoliters  =  1000  liters. 
1  liter  of  water  at  4  degrees  centigrade  weighs  1  kilogram. 
1  kilogram  =  1000  grams. 

1  meter  =  3.280899  feet  =  39.37079  inches. 
1  foot  =  0.304794  meter  =  30.4794  centimeters. 
1  inch  =  25.3995  millimeters  =  2.53995  centimeters. 

=  0.253995  decimeter  =  0.0253995  meter. 
1  sq.  meter  =  10.7643  sq.  feet  =  1550  sq.  inches. 
1  sq.  foot  =  0.0928997  sq.  meter  =  928.997  sq.  centimeters. 
1  sq.  inch  =  6.451368  sq.  centimeters. 
1  cu.  meter  =  35.316585  cu.  feet  =  264.1863  gallons. 

1  liter  =  0.035316585  cu.  feet;  =  0.2641863  gallons. 
1  cu.  foot  =  0.0283153  cu.  meters,  =  28.3153  liters. 
1  cu.  inch  =  0.0163861  liters,  =  16.38618  cu.  centimeters. 

1  gallon  =  3.7852  liters. 
1  liter  weighs  2.204672  English  pounds. 
1  cu.  foot  weighs  62.425  English  pounds. 
1  gallon  weighs  8.3448  English  pounds. 
1  gallon  =  231  cubic  inches. 

The  pressure  of  water  in  kilograms  is  equal 

per  square  meter         to  1000  h  (h  in  meters) 
"       "       decimeter    "       10  h 
centimeter  "       0.1  h 
"       "       millimeter  "  0.001  h. 
A  pressure  of  one  pound  per  square  inch  is  equal  to 

a  pressure  of  0.07031      kilo  per  square  centimeter 
"  0.0007031     "     "       "       millimeter. 

The  tensile,  shearing,  or  compressive  strength  of  any  material 
in  pounds  per  square  inch  multiplied  by  0.0007031  gives  the 
value  in  kilos  per  square  millimeter  and  multiplied  by  0.07031, 
the  value  in  kilos  per  square  centimeter. 

A  pressure  of  1  atmosphere  =  14.7  pounds  per  square  inch 
corresponds  to  a  pressure  of  1.03296  kilos  per  square  centimeter 
or  a  head  of  10.3296  meters.  2  g  =  19.61. 


210 


THE   FLOW   OF   WATER 


Thickness  of  walls  of  conduits : 

PD 


t 


m 


c, 


t,  D,  C  and  m  in  millimeters. 

P  in  kilos  per  square  millimeter  =  0.001  h. 


Material. 

m 

C 

Cast  iron 

2  8 

7  6 

Wrought  iron 

12  0 

1  5 

Steel         

14  0 

Lead     ,    

0  3 

7  6 

APPENDIX    III. 


Greatest  Efficiency  of  a  Conduit  of  a  Given  Diameter  as  a  Transmitter 

of  Energy. 

Most  Economical  Diameter  of  a  Conduit  Transmitting  Energy 
under  Pressure. 

I. 

IN  a  preceding  chapter  the  ratio  between  the  total  head  and 
the  head  lost  in  overcoming  frictional  resistances,  which  for  a 
conduit  of  a  given  diameter  under  a  given  head  corresponds  to  a 
maximum  of  efficiency,  has  been  mentioned. 

The  potential  energy  of  Qf3  of  water  delivered  per  second  at 
a  vertical  distance  H  above  the  generator  is  equal  to 

Q  62.4  H  foot-pounds, 
or  0.1134  QH  horsepowers. 

The  discharge  of  a  steel-riveted  conduit  in  /3  per  second  is  equal 
to 

Q  =  40  <P'7S&, 

which  gives  for  the  loss  of  head, 

H  = 

quently  the  net 
equal  to 


1062  d5'1 

Consequently  the  net  energy  transmitted  to  the  generator  is 
ual  to 


This  is  to  be  a  maximum. 

Regarding  Q  as  the  variable  and  equating  the  first  differential 
coefficient  to  zero  we  have 


211 


212  THE   FLOW   OF   WATER 

26     ^L  0.1134 


niio/i  -a 
hence  0.1134  H 


Q 


The  root  of  this  equation  corresponds  to  a  maximum.     We 
have  consequently,  for  the  state  of  maximum  efficiency 


26        1062  d5-1 

9  77 
or,   —  is  the  head  sacrificed  in  overcoming  frictional  resistances 

when  the  conduit  is  in  a  state  of  maximum  efficiency  as  a  trans- 
mitter of  energy. 

9  H 

We  have  also  for  the  discharge  which  corresponds  to  -  > 

26 


=  40  d2-1  0.57  S. 

Hence  the  efficiency  of  the  conduit  is  greatest  when  the  velocity 
and  the  discharge  are  0.57  times  the  velocity  and  discharge 
corresponding  to  the  total  head  H. 

II. 

Of  much  greater  importance  is  the  quest  after  the  most  eco- 
nomical diameter  of  a  conduit  for  a  given  discharge  and  under  a 
given  head,  a  subject  recently  investigated  by  A.  L.  Adams. 

The  function  of  a  pressure  pipe  is  the  transmission  of  energy 
with  a  minimum  of  loss;  the  usefulness  of  a  power  plant  as  a 
whole  depends  on  several  factors,  chief  amongst  which  is  the 
amount  of  revenue  derived  from  its  operation. 

In  comparison  with  the  power  transmitted  the  cost  of  a  con- 
duit transmitting  all  or  nearly  all  the  energy  would  be  exces- 
sive. The  conduit  having  a  diameter  just  sufficient  to  carry  the 
given  quantity  of  water  under  the  given  head  delivers  but  a 
small  percentage  of  the  gross  energy  and  its  cost  per  horsepower 
transmitted  is  equally  excessive  as  the  cost  of  the  conduit  deliver- 
ing all  the  energy.  The  diameter  of  a  conduit  just  sufficient  to 
carry  a  given  quantity  under  a  given  head  is  equal  to 


APPENDIX  III  213 

The  diameter  necessary  to  carry  the  same  quantity  with  a  loss 
of  TsW  °f  the  gross  energy  is  equal  to 


=  (1000)**  =  3.875  times   the   diameter, 

just  sufficient  to  carry  the  given  quantity. 

A  quantity  of  100  f3  of  water  delivered  at  an  elevation  of  1000 
feet  above  the  generator  possesses  a  potential  energy  of 

100  X  1000  X  0.1134  =  11,340  H.P. 

The  diameter  of  a  vertical  steel-riveted  conduit  just  sufficient 
to  carry  the  given  quantity, 

d  =  /I09\27=  1.404  feet. 
\40/ 

The  velocity  corresponding  to  this  diameter  is  equal  to 
V  =  50.8  X  (1.404)0-7  =  64.38  feet  per  second. 
The  energy  transmitted  is 

(64-38)2  X  0.1134  X  100  -  729.9  H.P. 

2# 

This  is  6.43  per  cent  of  the  gross  energy.  The  percentage 
transmitted  by  the  conduit  just  sufficient  is  not  constant  but 
decreases  with  decreasing  quantities  and  slopes. 

For  Q  =  10,  H  =  100,  L  =  1000,  for  instance,  the  gross  energy 
is  113.4  H.P.  and  the  energy  transmitted  3.635  H.P.,  which 
is  3.21  per  cent. 

The  diameter  corresponding  to  a  loss  of  ToV<i  of  the  gross  energy 
is,  for  a  vertical  steel-riveted  pipe  carrying  100  /3, 

3.875  X  1.404  =  5.437  feet. 
A  diameter  of  5.407  feet  transmits 

11  340  _  (100)  ¥  X  1000  X  0.1134 
1062  X  (5.437)5-1 

=  11,328.7  H.P. 

The  efficiency  of  the  two  conduits  of  1.404  and  5.437  feet 
diameter  is  consequently  as  729.9  to  11,328.7  or  1  to  15.52. 


214  THE  FLOW  OF   WATER 

At  n  dollars  per  H.P.  the  value  of  the  energy  transmitted  is 
equal  to 

D-0.n3*QHn-<&^. 

The  thickness  of  the  shell  of  riveted  pipes  is  made  equal  to 


Hence  the  cubic  contents  of  a  shell  one  foot  long 

,3  =  td  12  n 
1728   ' 

and  its  weight  (specific  gravity  7.854)  per  foot 
=  0.434  hd2  12  7i  490 
20,000       1728    : 
which  reduces  to 

w  =  0.0334  hd2  for  d  in  feet. 

The  weights  of  finished  pipes  indicate  that  the  additional 
weight  due  to  rivets,  laps  and  straps  is  sensibly  equal  to 

w  =  0.00607  hd2, 

so  that  the  total  weight  of  a  finished  pipe  amounts  to 
w  =  0.03947  hd*. 

At  m  dollars  a  pound  for  steel  the  cost  of  a  finished  pipe  will 

be 

Dl  =  0.03947  hd2m. 

If  we  now  compare  the  cost  of  the  two  conduits  of  1.404  and 
5.437  feet  with  the  value  of  the  power  lost  and  the  respective 
cost  of  the  pipes  per  horsepower  delivered  we  find,  taking  n  =  100 
and  m  =  0.06, 


d                                              

1.404  ft. 

5.437  ft 

Cost                     

2318 

34,803 

Value  of  energy  lost  
Cost  of  pipe  per  h.p  

1,061,000 
3.171 

1134 
3.072 

Between  the  two  extremes,  the  conduit  delivering  but  a  small 
percentage  of  the  gross  energy,  and  the  conduit  transmitting 
nearly  the  whole,  both  delivering  the  energy  at  an  equally  high 
expense  per  horsepower  transmitted,  there  is  evidently  a  con- 


APPENDIX  III 


215 


dition  more  favorable  to  economy  and  it  is  evident  that  the 
greatest  economy  exists  when 

L      Cost  of  Conduit _  a  minimum_ 

Value  of  energy  transmitted 

II.    Value  of  Energy  lost  +  Cost  of  conduit  =  a  minimum. 

The  value  of  the  energy  lost  +  cost  of  conduit  is 

0.1134  Q^Ln 
1062      d6-1 


0.03947  hd2Lm. 


Equating  the   first  differential  coefficient  with  regard  to  d  to 
zero  we  have 

5.1  X  0.1134  Q^Ln 

1062  d6*1 
which  gives 

5.1  X  0.1134Qyn      \" 
2  X  1062  X  0.03947  mh) 


+  2  X  0.03947  hdLm  =  0, 


1 
\ 


0.4962 


(Fl) 


Values  of  Q°-407,  H     and  n     are  found  in  the  table  below. 


For  steel  at  c  cents  per   pound  values  of     '   lo     and  0.4962 

f  —  ]    are  as  follows: 
\m  I 


c. 

0.4962 

/100\$i 

n  4Qfio    i                      • 

14     * 
m** 

2  \   m  ) 

5.0 

0.7566 

1.447 

5.5 

0.7463 

1.428 

6.0 

0.7381 

1.410 

6.5 

0.7291 

1.395 

7.0 

0.7213 

1.381 

7.5 

0.7146 

1.367 

8.0 

0.7079 

1.354 

If  we  again  take  the  previous  example  of  the  vertical  conduit 
1000  feet  long  and  carrying  100  /3  per  second,  we  find  for  n  = 
100,  m  =  0.06  and  H  a  mean  value  of  500,  from  the  tables, 

6.516 


1.410 


=  3.831  feet 


2.40 
46  inches;  very  near. 


216 


THE  FLOW  OF  WATER 


The  loss  of  energy  corresponding  to  this  diameter  is 


H.P.  = 


/      IPO      V 

\4Q  (3.831)2'7/ 
67.8 


L  0.1134  X  100 


and  the  net  horsepower  =  11,340  -  69.8  =  11,272.2.     The  three 
conduits  we  have  taken  as  examples  compare  as  follows: 


Cost  of  Con- 

Cost of 

d  in 
Feet. 

Cost  of 
Conduit. 

Value  of 
Energy  Lost. 

duit  +  Value 
of  Energy 

Conduit  per 
net 

Efficiency. 

Lost. 

Horsepower. 

1.404 

2,318 

1,061,000 

1,063,318 

3.171 

1.0 

3.831 

17,781 

6,380 

24,161 

1.574 

15.45 

5.437 

34,803 

1,134 

35,937 

3.072 

15.52 

Formula  I  as  given  above  gives  best  results  when  applied  to 
conduits  of  riveted  steel  under  high  pressures. 

The  experiments  of  Darcy,  Hamilton,  Smith,  C.  Hershel  indicate 
that  for  riveted  pipes  up  to  4  feet  in  diameter  the  mean  value  of 
the  coefficient  c  corresponding  to  a  velocity  of  one  foot  per  second 
is  equal  to  101.1,  or  nearly  so.  Taking  the  value  of  the  coefficient 
of  variation  of  c  equal  to  a  —  F°  the  exponential  equation 
corresponding  to  the  given  values  of  c  and  a  reads, 

V  =  63.66 
Q  = 


1622  d? 
This  gives  for  the  most  economical  diameter 


=  0.4465 


1  /2. 


m 


(FII) 


For  steel  at  c  cents  a  pound  and  n  =  100  dollars  the  values  of 
are  as  follows: 


0.4465  [- 

\m 


c=  0.4465  [- 

5.0=  1.371 
5.5  =  1.351 
6.0=  1.334 


6.5  =  1.319 

7.0  =  1.304 
7.5  =  1.291 
8.0=  1.279 


APPENDIX  III 


217 


Formula  II  is  best  suited  to  small  quantities  of  discharge  and 
.low  heads.  For  our  previous  example,  Q  =  100,  H  =  0.5  X  1000, 
the  formula  gives  d  =  3.797,  hence  0.034  feet  or  0.408  inch  less 
than  Formula  I.  The  difference  increases  with  the  decrease  of  the 
quantity;  for  small  diameters  the  difference  amounts  to  as  much 
as  one  inch.  It  will  be  observed  that  according  to  Formula  I, 
the  value  of  the  energy  lost  is  equal  to  f  f  =  0.392,  the  cost  of  the 
pipe.  Formula  II  gives  if  =  0.418. 


VALUES  OF  Q  AND  Q0-407. 


Q 

gO.407 

Q 

Q0.407 

Q 

gO.407 

Q 

Q0.407 

Q 

Q0.407 

0.5 

0.737 

15.5 

3.051 

36 

4.300 

82 

6.011 

210 

8.813 

1.0 

1.000 

16.0 

3.091 

37 

4.348 

84 

6.071 

220 

8.982 

1.5 

.179 

16.5 

3.130 

38 

4.395 

86 

6.128 

230 

9.146 

2.0 

.326 

17.0 

3.168 

39 

4.442 

88 

6.185 

240 

9.305 

2.5 

.452 

17.5 

3.206 

40 

4.488 

90 

6.243 

250 

9.462 

3.0 

.564 

18.0 

3.242 

41 

4.533 

92 

6.299 

260 

9.614 

3.5 

.665 

18.5 

3.279 

42 

4.578 

94 

6.354 

270 

9.763 

4.0 

.758 

19.0 

3.315 

43 

4.622 

96 

6.408 

280 

9.907 

4.5 

.844 

19.5 

3.350 

44 

4.665 

98 

6.463 

290 

10.063 

5.0 

.912 

20.0 

3.385 

45 

4.708 

100 

6.516 

300 

10.190 

5.5 

2.001 

20.5 

3.419 

46 

4.751 

105 

6.647 

310 

10.327 

6.0 

2.072 

21.0 

3.453 

47 

4.792 

110 

6.772 

320 

10.413 

6.5 

2.142 

21.5 

3.486 

48 

4.834 

115 

6.898 

330 

10.594 

7.0 

2.202 

22.0 

3.519 

49 

4.874 

120 

7.018 

340 

10.723 

7.5 

2.211 

22.5 

3.551 

50 

4.914 

125 

7.136 

350 

10.850 

8.0 

2.331 

23.0 

3.575 

52 

4.994 

130 

7.251 

360 

10.978 

8.5 

2.389 

23.5 

3.614 

54 

5.098 

135 

7.363 

370 

11.098 

9.0 

2.446 

24.0 

3.645 

56 

5.146 

140 

7.472 

380 

11.219 

9.5 

2.500 

24.5 

3.676 

58 

5.221 

145 

7.580 

390 

11.339 

10.0 

2.553 

25.0 

3.707 

60 

5.293 

150 

7.684 

400 

11.455 

10.5 

2.604 

26 

•3.766 

62 

5.364 

155 

7.789 

410 

11.572 

11.0 

2.654 

27 

3.824 

64 

5.434 

160 

7.890 

420 

11.686 

11.5 

2.702 

28 

3.881 

66 

5.502 

165 

7.990 

430 

11.799 

12.0 

2.759 

29 

3.937 

68 

5.570 

170 

8.087 

440 

11.909 

12.5 

2.795 

30 

3.992 

70 

5.636 

175 

8.184 

450 

12.018 

13.0 

2.840 

31 

4.046 

72 

5.701 

180 

8.277 

460 

12.127 

13.5 

2.885 

32 

4.098 

74 

5.765 

185 

8.370 

470 

12.223 

14.0 

2.929 

33 

4.150 

76 

5.828 

190 

8.463 

480 

12.388 

14.5 

2.969 

34 

4.201 

78 

5.887 

195 

8.551 

490 

12.443 

15.0 

3.011 

35 

4.251 

80 

5.951 

200 

8.640 

500 

12.545 

218 


THE  FLOW  OF  WATER 


VALUES  OF  H  AND 


N  AND 


H 

nH 

H 

ffH 

H 

Htt 

H 

H** 

H 

H^ 

5 

1.254 

95 

1.899 

270 

2.200 

500 

2.400 

860 

2.589 

10 

1.383 

100 

1.913 

280 

2.211 

520 

2.413 

880 

2.598 

15 

1.464 

110 

1.939 

290 

2.222 

540 

2.426 

900 

2.606 

20 

.525 

120 

1.966 

300 

2.233 

560 

2.438 

920 

2.615 

25 

.574 

130 

1.985 

310 

2.243 

580 

2.456 

940 

2.623 

30 

.614 

140 

2.006 

320 

2.253 

600 

2.462 

960 

2.631 

35 

.650 

150 

2.025 

330 

2.263 

620 

2.473 

980 

2.638 

40 

.681 

160 

2.044 

340 

2.273 

640 

2.485 

1000 

2.646 

45 

.709 

170 

2.061 

350 

2.282 

660 

2.495 

1050 

2.664 

50 

.735 

180 

2.098 

360 

2.291 

680 

2.506 

1100 

2.681 

55 

.758 

190 

2.094 

370 

2.300 

700 

2.516 

1150 

2.698 

60 

.780 

200 

2.109 

380 

2.309 

720 

2.526 

1200 

2.717 

65 

.800 

210 

2.123 

390 

2.319 

740 

2.536 

1250 

2.730 

70 

.819 

220 

2.138 

400 

2.325 

760 

2.545 

1300 

2.745 

75 

1.837 

230 

2.151 

420 

2.341 

780 

2.555 

1350 

2.760 

80 

1.854 

240 

2.164 

440 

2.357 

800 

2.564 

1400 

2.774 

85 

1.870 

250 

2.179 

460 

2.371 

820 

2.573 

1450 

2.788 

90 

1.885 

260 

2.189 

480 

2.396 

840 

2.581 

1500 

2.801 

III. 

CONDUITS  OF   PLANED  STAVES. 

Circular  conduits  of  planed  staves  are  occasionally  used  for 
heads  up  to  200  feet.  The  thickness  of  the  shell  of  such  con- 
duits is  usually  made  equal  to  2  or  2.5  inches.  For  these  dimen- 
sions a  conduit  one  foot  in  internal  diameter  will  contain  9  or  12 
feet  board  measure.  Owing  to  the  constant  addition  of  4  or  5 
inches  to  the  internal  diameter  the  number  of  feet  board  measure 
does  not  increase  with  d  but  with  cP,  very  near.  At  I  dollars  a 
foot  board  measure  the  cost  of  the  wooden  shell  put  in  place  will 
be  equal  to 

9  or  12  d^Ll  respectively. 

Likewise  the  length  of  the  tension  rods,  usually  f-inch  steel, 
increases  with  d^;  their  weight  is  therefore  proportional  to  dS . 
It  is  safe  to  allow  a  stress  of  15,000  pounds  per  square  inch  in  these 
rods  and  as  the  outside  diameter  of  the  one  foot  pipe  is  equal  to 
1.333  or  1.416  the  inside  diameter;  the  cost  of  the  metal  will  be 
equal,  at  ra  dollars  a  pound,  to 

0.0338  or  0.0359  ft  hLm. 


APPENDIX   III  219 

For  pipes  of  planed  staves  a  mean  value  of  the  coefficient  c 
corresponding  to  a  velocity  of  1  foot  per  second  is  equal  to  108,  or 
nearly  so.  Taking  the  coefficient  of  variation  of  c  equal  to  Vi8 
this  gives  the  exponential  equation 

Q  =  53.63 
H 

"     = 


48 

1848  d  g 
Value  of  power  lost  at  n  dollars  per  horsepower, 


1848  ( 

Equating  the  first  differential  coefficient  of  value  of  power  lost 
plus  cost  of  pipe  to  zero  we  have  (for  t  =  2  inches) 


43  QL  0.1134  n  +  17^8  d*hLm+  ^  9cT*LJ  =  0; 

9       1848  d  * 


hence 


It  is  possible  to  solve  this  equation  by  Homer's,  or  some  other 
method  of  approximation.  Fairly  good  results  are  obtained  by 
taking  a  mean  of  the  exponents  of  d. 

The  formula  will  then  read,  after  reduction, 

(Fill) 

for  t  =  2  inches.  For  t  =  2.5  inches  0.0678  is  substituted  for 
0.0638  and  10.8  I  for  8  /.  This  formula  gives  results  sometimes 
above,  sometimes  below  the  true  value.  Where  great  accuracy 
is  desired,  the  value  of  d  obtained  from  the  formula  may  be 
tested  by  putting  its  ninth  root  into  the  expression  0.0638,  or 
0.0678  X60  hm  +  8,  or  10.8  X51,  and  increasing  or  diminishing 
the  value  of  X  till  this  expression  is  equal  in  value  to 

43  Q™  0.1134  n 
9  1848 

It  is  to  be  observed  that  for  this  class  of  conduits  the  ratio 
between  the  value  of  the  power  lost  and  the  cost  of  the  pipe  which 


220  THE  FLOW   OF   WATER 

corresponds  to  a  maximum  of  economy  is  not  the  same  as  for 
metal  conduits.  If  the  tension  rods  did  not  enter  the  problem  the 
ratio  would  be  as  8  to  43;  as  it  is  the  ratio  is  variable  but  usually 
in  the  neighborhood  of  11  or  12  to  43  or  0.25  to  0.28  to  1.0. 

IV. 
CONDUITS  LINED  WITH   PLAIN  OR  ARMORED  CONCRETE. 

Concrete,  plain  or  armored,  is  coming  more  and  more  into 
favor  as  a  material  forming  the  shells  of  conduits  of  all  descrip- 
tions. Over  metal  and  wood  this  substance  possesses  the  great 
advantage  not  to  be  subject  to  corrosion  and  decay;  it  is  prac- 
tically indestructible.  Experiments  have  brought  to  light  the 
fact  that  plain  concrete  conduits  under  internal  pressure  fail 
when  the  stress  in  the  material  reaches  168  pounds  per  square 
inch  or  nearly  so.  Under  external  pressures,  however,  they  fail 
only  when  the  stress  reaches  1500  pounds  per  square  inch  or 
nearly  so.  Plain  concrete  is  therefore  not  economical  where  inter- 
nal pressures  enter  the  problem.  But  the  material  may  be  used 
to  great  advantage  when  great  quantities  of  water  are  to  be 
delivered  under  low  heads. 

The  thickness  of  the  shell  of  plain  concrete  conduits  as  com- 
monly used  for  sewers  and  other  conduits  not  subject  to  internal 
pressures  is  usually  made  equal  to 

2  inches  for  d  =     1  foot. 

4  inches  for  d  =    3  feet. 

8  inches  for  d  =    9  feet. 
12  inches  for  d  =  18  feet. 

These  conduits  will  fail  when  the  1  foot  pipe  is  under  a  head 
of  127  feet  or  the  18  foot  pipe  under  a  head  of  43  feet. 

The  thickness  of  the  shell  in  inches  of  such  conduits  is  propor- 
tional to  2  d0-63,  the  cubic  contents  of  the  shell  to  0.611  d1-63/3. 
At  c  dollars  per  /3  of  concrete  put  into  place  the  cost  of  such  a 
conduit  will  consequently  be 

0.611  &MLc. 

Taking  the  coefficient  of  variation  of  c  equal  to  V1S  the  exponen- 
tial equations  which  apply  to  flow  in  conduits  lined  with  concrete 
are  as  follows: 
m  =  0.95,  conduits  smoothly  dressed  with  neat  cement, 

Q  =  54.3  d*Msft, 


APPENDIX  III  221 

m  =  0.83,  conduits  lined  with   cement  plaster,  1  part  cement, 
2  parts  sand;  plain  concrete  washed  with  neat  cement, 

Q  =  50.2  d2-67'S", 

m  =  0.57,  conduits  lined  with  plain  concrete, 
Q  =  41.2  d2-™S&. 

For  m  =  0.95  the  value  of  the  power  lost  plus  the  cost  of  the 
conduit  will  be  equal  to 


of  which  the  first  differential  coefficient  equated  to  zero 


The  most  economical  diameter  will  be  equal 
for  m  =  0.95  to  d  =  0.2967 


\c 

for  m  =  0.83  to  d  =  0.3044  Q™  (-} 

W 

for  m  =  0.57  to  d  =  0.3247 


It  is  to  be  observed  that  this  class  of  conduits  is  in  the  state 
of  greatest  economy  when  the  value  of  the  power  lost  is  equal 

(for  m  =  0.83)  to  -i^-  =  0.323  of  the  cost  of  the  conduit. 
5.043 

Concrete  beams  armored  with  1.75  to  2  per  cent  steel  fail  when 
the  modulus  of  rupture  equals  2400  pounds  per  square  inch  or 
.  nearly  so.  Taking  10  as  a  factor  of  safety  the  working  stress 
for  internal  pressures  will  be  240  pounds  per  square  inch  and  the 
thickness  of  the  shell  will  be  equal  to 

_  0.434  hd 
480 

z  being  equal  to  1  inch  for  h  =  1  to  h  =  100,  and  vanishing  for 
h  =  1000.  Accordingly  the  thickness  of  the  shell  of  a  conduit 
1  foot  in  internal  diameter  for  h  =  1000  will  be  10.4  inches  and  the 
cubic  contents  of  the  shell  for  any  diameter  and  any  head  will  be 

(0.611  +  0.0035  h)  d1'63/8. 

fc 


222  THE  FLOW  OF  WATER 

Cracks  in  armored  concrete  begin  to  appear  when  the  stress 
in  the  steel  equals  12  to  15,000  pounds  per  square  inch.  A  safe 
working  stress  will  therefore  be  10,000  pounds  per  square  inch. 
Allowing  one-sixth  for  the  increase  of  the  diameter  where  the 
armoring  is  placed  in  the  1  foot  pipe  and  also  one-sixth  for  the 
laps  of  the  armoring,  the  weight  of  the  metal  in  the  1  foot  pipe 
will  be  for  h  =  1  equal  to  0.0445  pounds. 

But  the  thickness  of  the  shell  increases  with  h  and  conse- 
quently the  length  of  the  circumference  where  the  armoring  is 
placed.  The  necessary  increase  in  the  amount  of  the  armoring 
is  proportional  to  l{/h  very  near,  so  that  for  any  head  and  any 
diameter  the  weight  of  the  armoring  will  be 

0.0445  h&  d1-63  L. 

Using  these  values  we  find  for  the  most  economical  diameter, 
m  =  0.95, 


d      / 0.0003031  Q*n \™ , 

\0.0445  h&m  +  (0.611  +  0.0035  h)  J 

d  =  I 0.0003523  Q™n       *         \^ 

\0.0445  h    m+  (0.611  +  0.0035  h)  J 

A      I  0.000514    Q™n  \* 

(J^     =    I    —      •        ......  .  .  .  .    • 1        • 

V0.0445  h&m  +  (0.611  +  0.0035  h)  d 


m  =  0.57, 


In  these  equations  n  =  value  of  1  horsepower, 

m  =  value  of  1  pound  of  steel, 
c  =  value  of  1  cubic  foot  of  concrete. 


APPENDIX  in  223 

V. 

THE  MOST  ECONOMICAL  DIAMETER  FOR  METRIC  MEASURE. 
The  exponential  equation  for  steel  riveted  pipes  reads, 
Q  =  28  d?'7S". 
H  * 


541. 

Value  of  power  lost  at  n  dollars  per  kilowatt 

9.81  Q™Ln 
541.  4  d5-1  ' 

Allowing  a  tension  of   14  kgm.  per  square   millimeter  in  the 
steel  the  weights  of  the  shell  in  kilograms  will  be 

3.1416^7854  -  1.78246  «, 

and  allowing  for  rivets,  laps  and  straps,  the  cost  of  the  conduit 
at  ra  dollars  per  kilogram  will  be  equal  to 

2.083  d2hm, 
which  gives  for  the  most  economical  diameter 


,  =  /      5.1  X  9.81  Q¥n      \" 
V541.4  X  2  X  2.083  hm) 


A  mean  value  of  the  coefficient  c  corresponding  to  a  velocity 
of  1  meter  per  second  found  from  data  relating  to  flow  in  17 
riveted  conduits  including  the  largest  as  well  as  the  smallest  is 
equal  to  61.93.  Taking  a  =  V18  this  corresponds  to  the  exponen- 
tial equation, 

Q  =  29.76  d&S  &, 
from  which  we  find  for  the  most  economical  diameter 

d  =  0.5551  Q**  (^ 


In  these  equations  d  and  h  are  in  meters, 
Q  in  ra3  per  second, 
n  the  value  of  1  kilowatt, 
ra  the  value  of  1  kilogram  of  steel. 


INDEX. 


PAGE 

Alexander's  experiments 30 

Authorities  of  experimental  data 77 

Bazin  formula 168 

Bourdon  gauge 184 

Brass  tubes 72 

Channels  in  earth 47 

Channels,  proportions  of 117 

Circular  conduits 73 

Coefficient  C 15 

primary  determination 17 

variation  of 19,  196 

Coefficient  of  friction 71 

Conduits,  circular 73 

classification  according  to  coefficient  a 32 

forms  of  sections 113 

greatest  efficiency 211 

long,  circular 121 

masonry 115 

planed  staves 218 

open 43,  129 

discharge  of 188 

powers  of  diameters 145 

quantities  of  discharge 155 

riveted 59 

Coulomb's  investigations 31 

Cross-section  most  favorable  to  flow 129 

Current  meters. 190 

Curve,  friction  of 56 

Darcy  and  Hamilton  Smith's  experiments 34,  216 

Darcy  gauge 192 

Depths  of  water,  powers 147 

Determination  of  coefficient  C 17 

Diameters  of  velocities,  general  relations 121 

Diameter  of  conduits,  most  economical 211 

for  metric  measure 223 

powers  of 145 

Direction  current  meter 191 

Discharge  of  conduits 155,  184,  188 

225 


226  INDEX 

PAGE 

Discharge,  quantities,  of  semisquare 163 

weir 167 

Distribution,  energy 14 

head ,  .  .  .  12 

Double  float 189 

Energy,  distribution  of 14 

English  and  metric  equivalents 209 

Erosion,  resistance  to 48 

Exponential  equations 121 

metric  measure 205 

Fall,  primary  laws 4 

Floats 189 

Flow,  velocities  in  semisquare 159 

velocity  of 24 

Fluid  friction,  primary  laws. 8 

Formulae,  metric  measure 205 

practical  applications 62 

Francis'  formula 167 

Friction  in  curves 56 

Galvanized  iron  tubes 72 

Ganguillet  and  Kutter's  formula 203 

Harlacher  meter 191 

Head,  distribution  of 12 

loss  of 183 

Hubbel  and  Fenkell's  experiments 29 

Kinetic  energy 68 

Lawrence  and  Braunworth's  experiments 187 

Loss  of  head 183 

Masonry  conduits,  forms  of  sections 115 

Mean  hydraulic  radius,  relation  to  wet  perimeter 113 

Mean  hydraulic  radii,  roots  of 74 

Measurement,  loss  of  head 183 

Meters 191 

Metric  equivalents 209 

Metric  measure,  formulae  and  equations 205 

most  economical  diameter 223 

Notation iv 

Open  conduits 43,  129 

Pipes,  welded 72 

Pitot  tube 192 

Powers  of  depth  of  water 147,  151 

Pressure,  primary  laws 4 


INDEX  227 

PAGE 

Price  current  meter 191 

Primary  laws,  fluid  friction 8 

pressure  and  fall 4 

Prony's  formula 31 

Quantities,  use  of  tables  of 136 

Resistance  due  to  entrances  and  elbows 57 

Ritchie-Haskell  current  meter 191 

Riveted  conduits 59 

Rod  floats 189 

Roughness,  degree  of 76 

Saph  and  Schoder's  experiments 30 

Sections,  areas  of 117 

Semisquare,  quantities  of  discharge 163 

velocities  of  discharge 159 

Sewers 118 

Sheet  iron  tubes 72 

Slopes,  table  of  sines  of 143 

Stearns  and  Fitzgerald's  experiments 35 

Subsurface  float 189 

Surface  float 189 

Surface  mean  and  bottom  velocities 193 

TABLES: 

I.  II.  Variation  of  coefficient  C 19 

III.  Experimental  data  showing  extent   of  variation   of  C  with  the 

velocity  of  flow 37 

IV.A.    Weisbach's  coefficients  for  resistance  due  to  entrances,  elbows, 

etc 57 

IV.  Friction  in  curves 56 

V.  Roots  of  velocities 70 

VI.  Values  of  66  (t/r  +  m) 71 

VI.A.  Welded  pipes 72 

VII.  Circular  conduits 73 

VILA.  Roots  of  mean  hydraulic  radii 74 

A.  Values  of  R  and  areas  of  sections  in  terms  of  radius 117 

B.  Proportions  of  channels  of  maximum  values  of  R 117 

C.  Sines  of  slopes  and  roots  of  sines  of  slopes 143 

D.  Powers  of  diameters  of  conduits 145 

E.  Powers  of  mean  hydraulic  radii 147 

F.  Form  of  section  most  favorable  to  flow 151 

G.  Quantities  of  discharge  of  conduits 155 

H.  Velocities  of  flow  in  semisquare 159 

I.  Quantities  of  discharge  of  semisquare 163 

K.  Values  of  3.33  H* 168 

L.a,  Value  of  constant  in  Bazin's  formula 170 

L.b,  Value  of  Q  in  Bazin's  formula 171 


228  INDEX 

TABLES  (continued).  PAGE 

VIII.  Values  of  coefficients  indicating  degree  of  roughness 76 

IX.  List  of  authorities  whose  experimental  data  are  given 77 

X.  Experimental  data ' 82 

Tables  of  velocities,  use  of 136 

Tachometer,  Woltman's 191 

Thread  and  mean  velocity 193 

Variation  of  coefficient  C,  extent  of 37 

with  slope 196 

Velocities,  roots  of 70 

surface,  mean  and  bottom 193 

tables  of 136 

Velocity,  discharge  and  depth  of  water,  relations  between 131 

Velocity  measurements 190 

Velocity  of  flow,  variation  of  coefficient  C 24 

Venturi  meter,  theory  of 185 

Weir  discharges 167 

formulae 173 

Weisbach's  coefficients  of  resistance 57 

Welded  pipes 72 

Wet  perimeter,  relation  to  mean  hydraulic  radius 113 

roughness  of 21 

Woltman's  tachometer .  .  191 


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Pocket  size,  flexible  leather,  elaborately  illustrated  with  an  extensive  index 
and  patent  thumb  index  tabs.     1636  pp $5.00 

FOX,  WM.,  and  THOMAS,  C.  W.,  M.E.  A  Practical  Course  in  Mechan- 
ical Drawing.  With  plates.  Third  edition,  revised.  12mo,  cloth.$1.25 

GANT,  L.  W.     Elements  of  Electric  Traction,  for  Motormen    and 

Others.     Illustrated.     217  pp.,  8vo,  cloth net  $2.50 

GEIKIE,   J.     Structural   and  Field  Geology,   for   Students   of   Pure 

and  Applied  Science.     With  figures,  diagrams,  and  half-tone  plates.     8vo, 
cloth net,  $4.00 

GILLMORE,  Q.  A.,  Gen.     Practical  Treatise  on  the  Construction  of 

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12mo,  cloth $2.00 


STANDARD    TEXT    BOOKS.  5 

GOODEVE,  T.  M.  A  Text-Book  on  the  Steam-Engine.  With  a 
Supplement  on  Gas-Engines.  Twelfth  edition,  enlarged.  143  illustra- 
tions. 12mo,  cloth $2.00 

GUNTHER,  C.  O.,  Prof.  Integration  by  Trigonometric  and  Imag- 
inary Substitution.  With  an  Introduction  by  J.  Burkitt  Webb.  Illus- 
trated. 12mo,  cloth net,  $1.25 

GUY,  A.  E.     Experiments  on  the  Flexure  of  Beams,  resulting  in  the 

Discovery  of  New  Laws  of  Failure  by  Buckling.  Reprinted  from  the 
"American  Machinist."  With  diagrams  and  folding  plates.  8vo,  clothj 
illustrated.  122  pp net,  $1.25 

HAEDER,  HERMAN,   C.   E.     A  Handbook   on  the   Steam-Engine. 

With  especial  reference  to  small  and  medium  sized  engines.  Third 
English  edition,  re-edited  by  the  author  from  the  second  German  edi- 
tion, and  translated  with  considerable  additions  and  alterations  by  H.  H. 
P.  Powles.  Nearly  1100  illustrations.  12mo,  cloth $3.00 

HALE,  W.  J.,  Prof.  (Univ.  of  Mich.)  Calculations  of  General  Chem- 
istry, with  Definitions,  Explanations,  and  Problems.  174  pp.  12mo, 
cloth net,  $1.00 

HALL,  WM.  S.,  Prof.  Elements  of  the  Differential  and  Integral 
Calculus.  Sixth  edition,  revised.  8vo,  cloth.  Illustrated net,  $2.25 

-  Descriptive  Geometry;  With  Numerous  Problems  and  Practical 

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trated problems.  Second  edition.  Two  vols.,  cloth net,  $3.50 

HALSEY,  F.  A.    Slide-Valve  Gears:  an  Explanation  of  the  Action  and 

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edition,  revised  and  enlarged.  12mo,  cloth $1.50 

HANCOCK,  HERBERT.     Text-Book  of  Mechanics  and  Hydrostatics. 

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HAWKESWORTH,  J.     Graphical  Handbook  for  Reinforced  Concrete 

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conditions  of  external  loading,  together  with  practical  examples  showing 
the  method  of  using  each  plate.  4to,  cloth net,  $2.50 

HAY,    A.     Alternating    Currents;     Their   Theory*    Generation,    and 

Transformation.     8vo,  cloth,  illustrated net,  $2.50 

Principles      of      Alternate-Current     Working.        12mo,    cloth, 

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net,  $2.50 

HECK,  R.  C.  H.,  Prof.  The  Steam-Engine.  Vol.  I.  The  Thermo- 
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Steam-Turbine.     8vo,  cloth.     Illustrated net,  $5.00 


6  STANDARD    TEXT   BOOKS. 

HERRMANN,  GUSTAV.     The  Graphical  Statics  of  Mechanism.     A 

Guide  for  the  Use  of  Machinists,  Architects,  and  Engineers;  and  also  a 
Text-Book  for  Technical  Schools.  Translated  and  annotated  by  A.  P. 
Smith,  M.E.  7  folding  plates.  Sixth  edition.  12mo,  cloth $2.00 

HIROI,  I.     Statically-Indeterminate  Stresses  in  Frames  Commonly 

Used  for  Bridges.  With  figures,  diagrams,  and  examples.  12mo,  cloth, 
illustrated net,  $2.00 

HOPKINS,    N.    MONROE,    Prof.     Experimental    Electrochemistry. 

Theoretically  and  Experimentally  Treated.     300  pp.,   8vo.     Illustrated. 

net,  $3.00 

HOUGHTON,  C.  E.     The  Elements  of  Mechanics  of  Materials.     A 

text  for  students  in  engineering  courses.  Illustrated.  194  pp.,  12mo, 
cloth • net,  $2'00 

HUTCHINSON,  R.  W.,  Jr.  Long  Distance  Electric  Power  Trans- 
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Transformation,  Transmission,  and  Distribution.  12mo,  cloth,  illustrated, 
345  pp net,  $3.00 

JAMIESON,  ANDREW,  C.  E.     A  Text-Book  on  Steam  and  Steam- 

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of  Science  and  Art,  City  and  Guilds  of  London  Institute,  and  other  Engineer- 
ing students.  Fifteenth  edition,  revised.  Illustrated.'  12mo,  cloth. 

$3.00 

Elementary  Manual  on  Steam,  and  the  Steam-Engine.  Spe- 
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of  London  Institute,  and  other  Elementary  Engineering  Students.  Twelfth 
edition.  12mo,  cloth $1.50 

JANNETTAZ,  EDWARD.     A  Guide  to  the  Determination  of  Rocks: 

being  an  Introduction  to  Lithology.  Translated  from  the  French  by  G. 
W.  Plympton,  Professor  of  Physical  Science  at  Brooklyn  Polytechnic 
Institute.  Second  edition,  revised.  12mo,  cloth $1.50 

JOHNSTON,  J.  F.  W.,  Prof.,  and  CAMERON,  Sir  CHARLES.    Elements 

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cloth $2.60 

KAPP,  GISBERT,  C.  E.     Electric  Transmission  of  Energy,  and  its 

Transformation,  Subdivision,  and  Distribution.  A  practical  handbook. 
Fourth  edition,  revised.  12mo,  cloth $3.50 

KELLER,    S.    S.,    Prof.     Mathematics    for    Engineering    Students 

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KEMP,  JAMES  FURMAN,  A.B.,  E.M.     A  Handbook  of  Rocks;  for 

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KERSHAW,  J.  B.  C.  Electrometallurgy.  Illustrated.  303  pp.  8vo, 
cloth net,  $2.00 

KLEIN,  J.  F.  Design  of  a  High-Speed  Steam-engine.  With  notes 
diagrams,  formulas,  and  tables.  Second  edition,  revised  and  en- 
larged. 8vo,  cloth.  Illustrated.  257  pp net,  $5.00 

KNIGHT,  A.  M.,  Lieut.-Com.,  U.S.N.  Modern  Seamanship.  Illus- 
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8vo,  cloth,  illustrated net,  $6.00 

Half  morrocco $7.50 

KOESTER,  F.     Steam-Electric  Power  Plants  and  their  Construction. 

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their  Economical  Construction  and  Operation.  473  pp.,  340  illustrations. 
<Svo,  cloth net,  $5.00 

KRAUCH,    C.,    Dr.     Testing    of    Chemical    Reagents    for    Purity. 

Authorized  translation  of  the  Third  Edition,  by  J.  A.  Williamson  and  L.  W. 
Dupre.  With  additions  and  emendations  by  the  author.  8vo,  cloth, 

net,  $3.00 

LAMBORN,  L.  L.  Cottonseed  Products.  A  Manual  of  the  Treatment 
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tables,  figures,  full-page  plates,  and  a  large  folding  map.  8vo,  cloth, 
illustrated net,  $3.00 

LANCHESTER,    F.     W.     Aerodynamics:      Constituting    the    First 

Volume  of  a  Complete  Work  on  Aerial  Flight.  With  Appendices  on  the 
Velocity  and  Momemtum  of  Sound  Waves,  on  the  Theory  of  Soaring  Flight, 
etc.  With  numerous  diagrams  and  half-tones.  Illustrated.  442  pp  ,  8vo^ 
cloth net,  $6.00 

LASSAR-COHN,  Dr.  An  Introduction  to  Modern  Scientific  Chem- 
istry, in  the  form  of  popular  lectures  suited  to  University  Extension  Students 
and  general  readers.  Translated  from  the  author's  corrected  proofs  for 
the  second  German  edition,  by  M.  M.  Pattison  Muir,  M.A.  12mo,  cloth. 
Illustrated $2.00 

LATTA,  M.  N.     Handbook  of  American  Gas-Engineering  Practice. 

With   diagrams    and   tables.     8vo,    cloth,    illustrated,   460   pp.. net,  $4.50 

LEEDS,  C.  C.  Mechanical  Drawing  for  Trade  Schools.  High  School 
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4to,  oblong  cloth net,  $1.25 

Mechanical  Drawing  for  Trade  Schools.     Machinery  Trades' 

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LIVERMORE,  V.  P.,  and  WILLIAMS,  J.  How  to  Become  a  Com- 
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Operating  a  Street  Railway  Motor  Car;  also  giving  details  how  to  over- 
come certain  defects.  Revised  edition,  entirely  rewritten  and  en- 
larged. 16mo,  cloth,  illustrated $1.00 

LODGE,  OLIVER  J.  Elementary  Mechanics,  including  Hydrostatics 
and  Pneumatics.  Revised  edition.  12mo,  cloth $1.50 


8  STANDARD   TEXT  BOOKS. 

LUCKE,  C.  E.  Gas  Engine  Design.  With  figures  and  diagrams. 
Second  edition,  revised.  8vo,  cloth,  illustrated net,  $3.00 

LUM.E,  G.,  Ph.D.  Technical  Chemists'  Handbook.  Tables  and 
methods  of  analysis  for  manufacturers  of  inorganic  chemical  products. 
283  pp.  12mo,  leather .net,  $3.50 

LUQUER,    LEA   McILVAINE,    Ph.D.     Minerals    in    Rock    Sections. 

The  Practical  Method  of  Identifying  Minerals  in  Rock  Sections  with  the 
Microscope.  Especially  arranged  for  Students  in  Technical  and  Scientific 
Schools.  Third  edition,  revised.  8vo,  cloth.  Illustrated net,  $1.50 

MASSIE,  W.   W.,   and  UNDERBILL,   C.   R.     Wireless  Telegraphy 

and  Telephony  Popularly  Explained.  With  a  special  article  by  Nikola 
Tesla.  76  pp.  28  illustrations.  12mo,  cloth , net,  $1.00 

MEUCK,  C.  W.,  Prof.  Dairy  Laboratory  Guide.  12mo,  cloth, 
illustrated net,  $1.25 

MERCK,   E.     Chemical   Reagents:     Their   Purity   and   Tests.      250 

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MILLER,  E.  H.  Quantitative  Analysis  for  Mining  Engineers.  Second 
edition,  revised.  8vo,  cloth net,  $1.50 

MINIFIE,  WM.     Mechanical  Drawing.     A  Text-Book  of  Geometrical 

Drawing  for  the  use  of  Mechanics  and  Schools,  in  which  the  Definitions  and 
Rules  of  Geometry  are  familiarly  explained ;  the  Practical  Problems  are  ar- 
ranged from  the  most  simple  to  the  more  complex,  and  in  their  description 
technicalities  are  avoided  as  much  as  possible.  With  illustrations  for  Draw- 
ing Plans,  Sections,  and  Elevations  of  Railways  and  Machinery ;  an  Introduc- 
tion to  Isometrical  Drawing,  and  an  Essay  on  Linear  Perspective  and 
Shadows.  Illustrated  with  over  200  diagrams  engraved  on  steel.  Tenth 
thousand.  With  an  appendix  on  the  Theory  and  Application  of  Colors. 
8vo,  cloth $4.00 

MINIFIE,  WM.  Geometrical  Drawing.  Abridged  from  the  Octavo 
Edition,  for  the  use  of  schools.  Illustrated  with  48  steel  plates.  Ninth 
edition.  12mo,  cloth $2.00 

MOSES,  ALFRED  J.,  and  PARSONS,  C.  L.     Elements  of  Mineralogy, 

Crystallography,  and  Blow-Pipe  Analysis,  from  a  Practical  Standpoint. 
336  illustrations.  Fourth  edition.  8vo,  cloth $2.50 

NASMITH,  JOSEPH.  The  Student's  Cotton  Spinning.  Thirteenth 
thousand,  revised  and  enlarged.  8vo,  cloth.  Illustrated $3.00 

NUGENT,  E.     Treatise  on  Optics;    or,  Light  and  Sight  theoretically 

and  practically  treated,  with  the  application  to  Fine  Art  and  Industrial 
Pursuits.  With  103  illustrations.  12mo,  cloth $1.50 

OLSEN,  Prof.  J.  C.     Text-Book  of  Quantitative  Chemical  Analysis 

by  Gravimetric,  Electrolytic,  Volumetric,  and  Gasometric  Methods.  With 
seventy-two  Laboratory  Exercises  giving  the  analysis  of  Pure  Salts,  Alloys, 
Minerals,  and  Technical  Products.  Fourth  edition,  revised  and  en- 
larged. 8vo,  cloth.  Illustrated.  513  pp net,  $4.00 

OLSSON,  A.     Motor    Control    as    Used    in    Connection  with   Turret 

Turning  and  Gun  Elevating.  (The  Ward  Leonard  System.)  Illustrated. 
8vo,  Pamphlet,  27  pp.  (U.  S.  Navy  Electrical  Series,  No.  1.) net,  $.50 


STANDARD    TEXT    BOOKS.  9 

OUDIN,  MAURICE  A.     Standard  Polyphase  Apparatus  and  Systems, 

With  many  diagrams  and  figures.     Sixth  edition,  thoroughly  revised. 

Fully  illustrated.     8vo,  cloth $3.00 

PALAZ,  A.,  Sc.D.  A  Treatise  on  Industrial  Photometry,  with  special 
application  to  Electric  Lighting.  Authorized  translation  from  the  French 
by  George  W.  Patterson,  Jr.  Second  edition,  revised.  8vo,  cloth. 
Illustrated $4.00 

PARSHALL,  H.  F.,  and  HOBART,  H.  M.     Armature  Windings   of 

Electric  Machines.     With  140  full-page  plates,  65  tables,  and  165  pages 
of  descriptive  letter-press.     Second  edition.     4to,  cloth $7.50 

Electric  Railway  Engineering.     With  numerous  tables,  figures, 

and  folding  plates.     4to,  cloth,  463  pp.,  illustrated net,  $10.00 

PAULDING,  CHAS,  P.     Practical  Laws  and  Data  on  Condensation 

of   Steam   in   Covered   and   Bare   Pipes.     12mo,    cloth.     Illustrated.     102 
pages net,  $2.00 

The  Transmission  of  Heat  through  Cold-Storage  Insulation. 


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Manual  for  Refrigerating  Engineers.  12mo,  cloth.  41  pp.  Illustrated. 

net,  $1.00 

PERRINE,  F.  A.  C.,  A.M.,  D.Sc.  Conductors  for  Electrical  Dis- 
tribution; Their  Manufacture  and  Materials,  the  Calculation  of  the  Cir- 
cuits, Pole  Line  Construction,  Underground  Working  and  other  Uses. 
With  diagrams  and  engravings.  Second  edition,  revised.  8vo, 
cloth net,  $3.50 

PERRY,   JOHN.     Applied  Mechanics.     A   Treatise   for  the   Use   of 

Students  who  have  time  to  work  experimental,  numerical,  and  graphical 
exercises  illustrating  the  subject.  New  edition,  revised  and  enlarged. 
650  pages.  8vo,  cloth net,  $2.50 

PLATTNER.     Manual  of  Qualitative  and  Quantitative  Analysis  with 

the  Blow-Pipe.  From  the  last  German  edition,  revised  and  enlarged, 
by  Prof.  Th.  Richter,  of  the  Royal  Saxon  Mining  Academy.  Translated 
by  Prof.  H.  B.  Cornwall,  assisted  by  John  H.  Caswell.  Illustrated  with 
78'woodcuts.  Eighth  edition,  revised.  463  pages.  8vo,  cloth,  .net,  $4.00 

POPE,  F.  L.  Modern  Practice  of  the  Electric  Telegraph.  A  Tech- 
nical Handbook  for  Electricians,  Managers,  and  Operators.  Seventeenth 
edition,  rewritten  and  enlarged,  and  fully  illustrated.  8vo,  cloth  .$1.50 

PRELINI,  CHARLES.  Tunneling.  A  Practical  Treatise  containing 
149  Working  Drawings  and  Figures.  With  additions  by  Charles  S.  Hill, 
C.E.,  Associate  Editor  "Engineering  News."  Third  edition,  revised. 
8vo,  cloth.  Illustrated $3.00 


—  Earth  and  Rock  Excavation.     A  Manual  for  Engineers,  Contractors, 
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10  STANDARD   TEXT  BOOKS. 

PRESCOTT,  A.  B.,  Prof.  Organic  Analysis.  A  Manual  of  the 
Descriptive  and  Analytical  Chemistry  of  Certain  Carbon  Compounds  in 
Common  Use;  a  Guide  in  the  Qualitative  and  Quantitative  Analysis  of 
Organic  Materials  in  Commercial  and  Pharmaceutical  Assays,  in  the  Esti- 
mation of  Impurities  under  Authorized  Standards,  and  in  Forensic  Exami- 
nations for  Poisons,  with  Directions  for  Elementary  Organic  Analysis. 
Sixth  edition.  8vo,  cloth $5.00 

and  SULLIVAN,  E.  C.     First  Book  in  Qualitative  Chemistry. 

Eleventh  edition.     12mo,  cloth net,  $1.50 

and  OTIS  COE  JOHNSON.     Qualitative  Chemical  Analysis.     A 

Guide  in  the  Practical  Study  of  Chemistry  and  in  the  Work  of  Analysis. 
Sixth  revised  and  enlarged  edition,  entirely  rewritten,  with  an 
Appendix  by  H.  H.  Willard  containing  a  few  improved  methods  of  Analysis. 
8vo,  cloth net,  $3.50 

RANKINE,  W.  J.  MACQUORN,  C.E.,  LL.D.,  F.R.S.     Machinery  and 

Mill-work.  Comprising  the  Geometry,  Motions,  Work,  Strength,  Con- 
struction, and  Objects  of  Machines,  etc.  Illustrated  with  nearly  300 
woodcuts.  Seventh  edition.  Thoroughly  revised  by  W.  J.  Millar.  8vo, 
cloth $5.00 

The  Steam-Engine  and  Other  Prune  Movers.    With  diagram  of 

the  Mechanical  Properties  of  Steam.  With  folding  plates,  numerous 
tables  and  illustrations.  Fifteenth  edition.  Thoroughly  revised  by  W.  J. 
Millar.  8vo,  cloth $5.00 

-Useful  Rules  and  Tables  for  Engineers  and  Others.  With 
appendix,  tables,  tests,  and  formulae  for  the  use  of  Electrical  Engineers. 
Comprising  Submarine  Electrical  Engineering,  Electric  Lighting,  and 
Transmission  of  Power.  By  Andrew  Jamieson,  C.E.,  F.R.S.E.  Seventh 
edition,  thoroughly  revised  by  W.  J.  Millar.  8vo,  cloth $4.00 

A  Mechanical  Text-Book.      By  Prof.  Macquorn  Rankine  and  E.  E. 

Bamber,  C.E.  With  numerous  illustrations.  Fifth  edition.  8vo, 
cloth $3.50 

RANKINE,  W.  J.  MACQUORN,  C.E.,  LL.D.,  F.R,S.  Applied  Me- 
chanics. Comprising  the  Principles  of  Statics  and  Cinematics,  and  Theory 
of  Structures,  Mechanics,  and  Machines.  With  numerous  diagrams. 
Eighteenth  edition.  Thoroughly  revised  by  W.J.Millar.  8vo,  cloth  $5.00 

Civil  Engineering,      Comprising   Engineering,    Surveys,    Earthwork, 

Foundations,  Masonry,  Carpentry,  Metal-Work,  Roads,  Railways,  Canals, 
Rivers,  Water-Works,  Harbors,  etc.  With  numerous  tables  and  illus- 
trations. Twenty-third  edition.  Thoroughly  revised  by  W.  J.  Millar. 
8vo,  cloth $6.50 

JIATEAU,    A.     Experimental    Researches    on    the    Flow    of    Steam 

Through  Nozzles  and  Orifices,  to  which  is  added  a  note  on  The  Flow  of 
Hot  Water.  Authorized  translation  by  H.  Boyd  Brydon.  12mo,  cloth. 
Illustrated -net,  $1.50 

RAUTENSTRAUCH,    W.,    Prof.,  and    WILLIAMS,    J.    T.     Machine 

Drafting  and  Empirical  Design.  A  Textbook  for  Students  in  Engineering 
Schools  and  Others  Who  are  Beginning  the  Study  of  Drawing  as  Applied 
to  Machine  Design.  Part  I.  Machine  Drafting.  Illustrated,  70  pp., 

8vo,  cloth net,  $1.25 

Complete  in  Two  Parts.     Part  II  in  preparation. 


STANDARD    TEXT  BOOKS.  11' 

RAYMOND,  E.  B.  Alternating  Current  Engineering  Practically 
Treated.  Third  edition,  revised  and  enlarged,  with  an  additional 
chapter  on  "The  Rotary  Converter."  12mo,  cloth.  Illustrated.  232  pages. 

net,  $2.50- 

REINHARDT,  CHAS.  W.     Lettering  for  Draughtsmen,  Engineers  and 

Students.  A  Practical  System  of  Free-hand  Lettering  for  Working  Draw- 
ings. New  and  revised  edition.  Thirty-first  thousand.  Oblong  boards. 

$1.00 

RICE,  J.  M.,  Prof.,  and  JOHNSON,  W.  W.,  Prof.  On  a  New  Method 
of  Obtaining  the  Differential  of  Functions,  with  especial  reference  to  the 
Newtonian  Conception  of  Rates  of  Velocities.  12mo,  paper $0.50 

RIPPER,  WILLIAM.     A  Course  of  Instruction  in  Machine  Drawing 

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WEISBACH,  JULIUS.  A  Manual  of  Theoretical  Mechanics.  Ninth 
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WINGHELL,   N.   H.,  and  A.   N.     Elements   of  Optical  Mineralogy. 

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